Scuola Normale Superiore di Pisa

Classe di Scienze Matematiche, Fisiche e Naturali

PhD Thesis

On the geometry of the space of probability measures in ℝ^{n} endowed with the quadratic

optimal transport distance

Nicola Gigli

Advisor

prof. Luigi Ambrosio

2002 – 2004

per mia sorella per mio fratello

e per me

Contents

Introduction vii

Chapter 1. Preliminary results 1 1. Transport of measures 1 2. Tightness, 2-boundedness and 2-uniform integrability 2 3. Dudley’s lemma 6 4. On the solution of the optimal transport problem 7

Chapter 2. The distance W and its geometry 11 1. The distance W 11 2. Interpolation and geodesics 14 3. The curvature of (𝒫_{2}(ℝ^{d}), W) 17 4. The weak topology of 𝒫_{2}(X) 20 5. Convergence of maps 22

Chapter 3. The regular tangent space 29 1. Continuity equations and AC curves 29 2. The regular tangent space 35

Chapter 4. The geometric tangent space 41 1. An introduction to Tan_{μ}(𝒫_{2}(ℝ^{d})) 41 2. Directional derivative of W^{2}(·, ν) 46 3. Tangent cone for P C spaces 48 4. On the inclusion Tan_{μ}(𝒫_{2}(ℝ^{d})) ↪ Tan_{μ}(𝒫_{2}(ℝ^{d})) 51 5. Operations on 𝒫_{2}(ℝ^{2d})_{μ} 54 6. The projection onto Tan_{μ}(𝒫_{2}(ℝ^{d})) 61

Chapter 5. Geodesically convex sets 73 1. Halfspaces 74 2. Strongly geodesically convex sets 75 3. Geodesically convex subsets of 𝒫_{2}^{r}(ℝ^{d}) 76 4. Subdifferentials and differentiability properties of W^{2} 82

Chapter 6. An approach to second order analysis 89 1. The case of a manifold embedded in ℝ^{d} 89 2. On the angle between tangent spaces in 𝒫_{2}(ℝ^{d}) 93 3. Regular curves 98 4. Parallel transport 106

vi CONTENTS

5. Covariant derivative and curvature operator 118 6. A distance on the tangent bundle 123

Bibliography 137

Introduction

In the last years, starting from the seminal papers [45, 16, 46, 15] the theory of optimal transportation and the geometry induced by the optimal transportation problem on the space of probability measures have received a lot of attention (see [36, 51, 52, 11]). Motivations for this interest come from application to Optimization, Functional and Geometric Inequalities, Evolution PDE’s, and, in even more recent times, Riemannian Geometry.

The aim of this thesis is to provide an comprehensive study of the geometry of the space of probability measures in ℝ^{d} endowed with the quadratic optimal transportation distance. Some of the results described here are contained in the book [11], written with L. Ambrosio and G. Savaré (those in Chapter 3 and part of those in Chapter 4), while other results are presented here for the first time (in Chapter 4, 5 and 6). In particular we complete the analysis performed in [11] of a “calculus” with transport plans (considering a larger tangent bundle which includes plans as well), showing that a nice Hilbertian structure can be extended up to this level. In addition, still having in mind an Hilbertian analogy, we study the closure with respect to weak topologies of geodesically convex sets in the space of probability measures, showing that one of the assumptions made in [11], in the theory of gradient flows for convex functionals, is redundant. Finally, we try to make a complete analysis (with positive results, and counterexamples as well) of the problem of parallel transport in these spaces, showing that the parallel transport is well-defined and unique along a class of regular curves.

Now we pass to a more detailed description of the thesis’s content.

In Chapter 1 we recall the measure-theoretic results we will need in the work. In particular, in the first section we study the definition and the first properties of the push-forward of measures: given μ ∈ 𝒫(ℝ^{d}) and a μ-measurable function f : ℝ^{d} → ℝ^{d′}, the push-forward f_{#}μ of μ through f is the probability measure on ℝ^{d′} defined by

f_{#}μ(A) := μ(f^{−1}(A)), ∀ Borel set A ⊂ ℝ^{d′}.

In the last section we shortly discuss the Kantorovich formulation of the quadratic optimal transportation problem, i.e. the problem of finding, given two measures μ, ν in 𝒫(ℝ^{d}), the minimum of

(0.1)  ∫ |x − y|^{2}dγ(x, y),

among the set Adm(μ, ν) of all admissible plans γ, i.e. the set of those probability measure γ on ℝ^{d} × ℝ^{d} such that π_{#}^{1}γ = μ, π_{#}^{2}γ = ν, π^{1}, π^{2} being the projections onto the first and the second coordinate, respectively. The aim is just to recall the cornerstones of the theory, without entering into the details of the proofs (for this, we refer for instance to [11] and to [51]). The key results are the existence of the minimum of the optimal transportation problem and Brenier’s theorem: the former ensures that for any couple of measures μ, ν in 𝒫(ℝ^{d}) a minimizing plan always exists, the latter says that if μ and ν satisfy ∫ |x|^{2}dμ, ∫ |x|^{2}dν < ∞ and μ is regular (i.e. μ(Σ) = 0 for any H^{d−1} rectifiable set Σ ⊂ ℝ^{d}), then the minimizing plan γ is unique and there exists a (unique μ-a.e.) convex function φ : ℝ^{d} → ℝ such that γ = (Id, ∇φ)_{#}μ, Id being the identity operator.

In Chapter 2 we begin the study of the distance W, recalling some well-known facts. In the first section we introduce the space of probability measures with bounded second moment

(0.2)  𝒫_{2}(ℝ^{d}) := {μ ∈ 𝒫(ℝ^{d}) : ∫ |x|^{2}dμ(x) < +∞},

and the distance W on 𝒫_{2}(ℝ^{d}):

(0.3)  W^{2}(μ, ν) := min { ∫ |x − y|^{2}dγ : γ ∈ Adm(μ, ν)}.

We call optimal a plan γ ∈ Adm(μ, ν) which attains the minimum in the above expression.

In the second section we show that the space (𝒫_{2}(ℝ^{d}), W) is a length space and we characterize its geodesics: we show that a curve t → μ_{t} is a constant speed geodesic from μ_{0} to μ_{1} with constant velocity W (μ_{0}, μ_{1}) if and only if there exists an optimal plan γ such that

(0.4)  μ_{t} = ((1 − t)π^{1} + tπ^{2})_{#}γ.

The interpolation between two measures μ_{0}, μ_{1} given by the above formula was first introduced by McCann in [45] to prove uniqueness of the minimum of a certain class of functionals in 𝒫(ℝ^{d}): for this purpose he studied the displacement convexity of those functionals (now also known as geodesic convexity on (𝒫_{2}(ℝ^{d}), W)), i.e. convexity along the curves t → μ_{t}, with μ_{t} defined as above.

In the third section we begin the study of the curvature of the space (𝒫_{2}(ℝ^{d}), W) by showing that it is positively curved in the sense of Aleksandrov, that is: for every constant speed geodesic μ_{t} and every measure σ it holds

(0.5)  W^{2}(μ_{t}, σ) ≥ (1 − t)W^{2}(μ_{0}, σ) + tW^{2}(μ_{1}, σ) − t(1 − t)W^{2}(μ_{0}, μ_{1}),

(observe that in the flat Euclidean setting the squared distance satisfies the equality in this equation). A way to interpret equation (0.5) is: the funtion W^{2}(·, σ) is (-1)–concave along constant speed geodesics. It is then interesting to know that the same function is not λ–convex for any λ: we show this fact with an explicit counterexample. In the fourth section we introduce the weak topology τ on 𝒫_{2}(ℝ^{d}), which will play a central role in Chapter 5, as the topology such that a sequence (μ_{n}) on 𝒫_{2}(ℝ^{d}) converges to μ iff

\begin{aligned}∫ φdμ_{n} & → ∫ φdμ ∀φ ∈ C_{c}(ℝ^{d}, ℝ), \\ ∫ |x|^{2}dμ_{n} & < C, ∀n ∈ ℕ. \end{aligned} ; (0.6)

The reason of such a definition is that, at least from a formal viewpoint, many properties of (𝒫_{2}(ℝ^{d}), W) are the natural generalization of properties valid in an Hilbert setting: in these cases the topology induced by W plays the role of the strong topology and τ plays the one of the weak topology (for example, it is possible to show that W is lower semicontinuous w.r.t. τ and that bounded sets are relatively compact w.r.t. τ). Moreover, recalling the first part of [11], it is worth noticing that the topology τ is the natural weak topology in the space 𝒫_{2}(ℝ^{d}) for the study, by compactness methods, of the minimizing movements scheme (the implict Euler scheme canonically used for the approximation of gradient flows). We do not investigate the application to PDE’s in this work and we just point out that the introduction of τ does not add new results to [11]: the definition of τ simply clarifies the ideas, and helps putting everything in the right perspective. The last section is devoted to the development of some useful tools concerning the convergence of maps T_{n} ∈ L_{μ_{n}}^{2} when the base measures μ_{n} may depend on n.

Chapter 3 is the key one to start the study of the Riemannian structure of (𝒫_{2}(ℝ^{d}), W), first studied in the pioneering work [46] and then formalized in [11]. In the first section we characterize absolutely continuous curves in 𝒫_{2}(ℝ^{d}) through the study of the continuity equation

(0.7)  \frac{d}{dt}μ_{t} + ∇ · (v_{t}μ_{t}) = 0.

Indeed we show that, for any absolutely continuous curve, for a.e. t there exists a vector v_{t} ∈ L^{2}(μ_{t}, ℝ^{d}) (L_{μ_{t}}^{2} in the sequel) such that (0.7) holds and the bound

(0.8)  ∥v_{t}∥_{μ_{t}} ≤ |μ̇_{t}|

is valid for a.e. t (where ∥v∥_{μ} is the norm of the vector v in L_{μ}^{2}). Conversely, every solution (μ_{t}, v_{t}) of (0.7) with

(0.9)  ∫_{0}^{1} ∥v_{t}∥_{μ_{t}}dt < +∞

gives a curve t → μ_{t} which is a.e. equal to an absolutely continuous curve for which |μ̇_{t}| ≤ ∥v_{t}∥_{μ_{t}} holds.

Having this characterization in mind, it is natural to think to the v_{t}’s as the “velocity” vectors of the curve μ_{t}. The only problem is that, for a given μ_{t}, there is more than one family of v_{t}’s which satisfies (0.7): indeed, it is sufficient to find vectors w_{t} such that ∇ · (w_{t}μ_{t}) = 0 to find a new family v_{t} + w_{t} still satisfying (0.7). So we need to understand which v_{t}’s we should think as the right ones. There are essentially two ways to answer to this question, both of them leading to the same result: on one hand, one can observe that by (0.7) the action of v_{t} is defined only on the gradient of smooth functions. On the other hand, one can observe that by the linearity of the expression ∇·(v_{t}μ_{t}) and the strict convexity of the L^{2} norm, there exists only one family of vectors v_{t} of minimal norm satisfying (0.7): such vectors are characterized by ∫ ⟨v_{t}, w_{t}⟩dμ_{t} = 0 for every w_{t} such that ∇ · (w_{t}μ_{t}) = 0 and satisfy, by what we said on the solution of (0.7), ∥v_{t}∥_{μ_{t}} = |μ̇_{t}| for a.e. t. In both cases we are lead to define the velocity vectors of μ_{t} as those satisfying (0.7) and belonging to the tangent space at μ_{t}, defined, for a general measure μ, as

(0.10)  Tan_{μ}(𝒫_{2}(ℝ^{d})) := \overline{\{∇φ : φ ∈ C_{c}^{∞}(ℝ^{d})\}}^{L_{μ}^{2}},

which we endow with the L_{μ}^{2} norm.

It should be noted carefully that, with this definition, the space (𝒫_{2}(ℝ^{d}), W) is not an infinite dimensional Riemannian manifold in the canonical sense: indeed, it can be shown that the natural exponential map Tan_{μ}(𝒫_{2}(ℝ^{d})) ∋ v → exp_{μ}(v) := (Id + v)_{#}μ has injectivity radius 0 for every μ. However, there are quite a lot analogies with that structure. The first one is that the following formula, known as Benamou-Brenier formula [15], holds:

(0.11)  W (μ_{0}, μ_{1}) = min \{ ∫_{0}^{1} ∥v_{t}∥_{μ_{t}}dt : \frac{d}{dt}μ_{t} + ∇ · (v_{t}μ_{t}) = 0\},

which shows that we can recover the distance between two measures by minimizing the length of the connecting curves.

We conclude the Chapter with a first study of the differential properties of W^{2}(·, σ), showing that this functional is differentiable at regular measures μ and that its differential is v := 2(Id − T_{μ}^{σ}) (where T_{μ}^{ν} is the unique optimal transport map from μ to ν). This means that the following equation holds:

(0.12)  lim_{ν→μ} \frac{W^{2}(ν, σ) − W^{2}(μ, σ) − ∫ ⟨v, T_{μ}^{ν} − Id⟩dμ}{W (μ, ν)} = 0.

In Chapter 4 we deeply analyze a topic introduced in the appendix of [11]: a more general tangent space which includes not only maps, as Tan_{μ}(𝒫_{2}(ℝ^{d})), but even plans. With the same terminology of [11], we call this space the geometric tangent space. The reason for the introduction of such a space is that the space Tan_{μ}(𝒫_{2}(ℝ^{d})) is not completely satisfactory when dealing with non-regular measures: for example if μ = δ_{x_{0}} is a Dirac mass, then Tan_{μ}(𝒫_{2}(ℝ^{d})) is isomorphic to ℝ^{d} and the image of the exponential map consists only of Dirac masses. If we want to define a tangent space with a surjective exponential map, we have to allow the “splitting” of masses, in the same spirit of Kantorovich relaxation, involving plans, of Monge’s problem, involving maps. In the first section of the Chapter we introduce the geometric tangent space Tan_{μ}(𝒫_{2}(ℝ^{d})) via an analogy with the case of a regular manifold M embedded in ℝ^{N}: for an embedded manifold M, one may say that a vector v belongs to the tangent space of M in x_{0} if and only if there exists a constant speed geodesic t → x_{t} such that v = lim_{t}(x_{t} − x_{0})/t, therefore the tangent space can be defined by studying only the behavior of geodesics. With the formal identification of points x_{t} of the manifold with measures μ_{t} in 𝒫_{2}(ℝ^{d}), of the vector x_{t} − x_{0} with the plan (π^{1}, π^{2} − π^{1})_{#}γ_{t}, where γ_{t} is an optimal plan between μ_{0} and μ_{t}, and of the rescaled vector (x_{t} − x_{0})/t with the “rescaled” plan (π^{1}, (π^{2} − π^{1})/t)_{#}γ_{t}, we know that our tangent space has to contain at least all the plans of the kind

lim_{t→0^{+}} (π^{1}, \frac{π^{2} − π^{1}}{t} )_{#} γ_{t},

where γ_{t} is any choice of optimal plans from a fixed measure μ_{0} and the point μ_{t} of a constant speed geodesic t → μ_{t}. Of course we have to define the topology of the limit. The characterization of geodesics given by equation (0.4) allows us to say that the previous limit is equal to (because eventually equal to) (π^{1}, (π^{2} − π^{1})/T)_{#}γ, where γ is an optimal plan and T > 0 is a positive number. Reversing the last formula we know that Tan_{μ}(𝒫_{2}(ℝ^{d})) has to contain at least the set Geodμ defined as

Geodμ := {γ ∈ 𝒫_{2}(ℝ^{2d})_{μ} : (π^{1}, π^{1} + T π^{2})_{#}γ is optimal for some T > 0},

where 𝒫_{2}(ℝ^{2d})_{μ} is the set of all plans in 𝒫_{2}(ℝ^{2d}) whose first marginal is μ. Given that we want the regular tangent space to be isometrically included in Tan_{μ}(𝒫_{2}(ℝ^{d})) via the natural embedding v → ℑ(v) := (Id, v)_{#}μ, we are lead to endow 𝒫_{2}(ℝ^{2d})_{μ} with the distance

W_{μ}^{2}(γ_{1}, γ_{2}) := ∫ W^{2}((γ_{1})_{x}, (γ_{2})_{x})dμ(x),

where γ_{x} is the disintegration of γ w.r.t. its first marginal. The definition of the geometric tangent space will then be

Tan_{μ}(𝒫_{2}(ℝ^{d})) := \overline{Geodμ}^{W_{μ}},

meaning that the closure is taken w.r.t. the distance W_{μ}. Given that

(0.13)  ∥f − g∥_{μ} = W_{μ}(ℑ(f), ℑ(g))

and that for every regular function φ the map Id + t∇φ is optimal for sufficiently small t, we have that Tan_{μ}(𝒫_{2}(ℝ^{d})) is naturally embedded in Tan_{μ}(𝒫_{2}(ℝ^{d})). Equation (0.13) suggests a way to define the norm of plan and the scalar product between plans as following:

(0.14a)  ∥γ∥^{2} := W_{μ}^{2}(γ, ℑ(0)) = ∫ |x_{2}|^{2}dγ(x_{1}, x_{2}),
(0.14b)  2⟨γ_{1}, γ_{2}⟩ := ∥γ_{1}∥^{2} + ∥γ_{2}∥^{2} − W_{μ}^{2}(γ_{1}, γ_{2}).

Furthermore, the exponential map γ → exp(γ) := (π^{1} + π^{2})_{#}γ is clearly surjective, and the inverse function exp_{μ}^{−1}(ν) is well defined by

exp_{μ}^{−1}(ν):={γ ∈Tan_{μ}(𝒫_{2}(ℝ^{d})) : (π^{1}, π^{1} + π^{2})_{#}γ is optimal between μ and ν},

(which is not the functional inverse of exp).

In the second section we derive a useful and interesting formula to compute the directional derivative of W^{2}(·, σ) along an interpolation curve t → μ_{t} := (π^{1}, π^{1} + tπ^{2})_{#}γ, where γ is any plan in 𝒫_{2}(ℝ^{2d}) not necessarily optimal. The formula is:

\frac{d}{dt}W^{2}(μ_{t}, σ)|_{t=0} = −2 max_{σ∈exp_{μ}^{−1}(σ)} ⟨γ, σ⟩,

which is formally equivalent to the one valid in Riemannian manifolds.

In the third section, following the same approach given in [11] (see also [4, 20, 49]), we introduce an abstract notion of tangent cone for positively curved length spaces (i.e. length spaces which satisfy an inequality like (0.5)), showing that a notion of angle between geodesics and of norm of a geodesic are well defined, and that in the case of (𝒫_{2}(ℝ^{d}), W) these notions are consistent with those defined in (0.14).

In the fourth section we conclude the study of the embedding

Tan_{μ}(𝒫_{2}(ℝ^{d})) ↪ Tan_{μ}(𝒫_{2}(ℝ^{d})).

by showing the non trivial fact that the subset of Tan_{μ}(𝒫_{2}(ℝ^{d})) of plans induced by maps is isomorphic through ℑ^{−1} to Tan_{μ}(𝒫_{2}(ℝ^{d})): this means that every optimal map is tangent, regardless of any assumption on the regularity of the base measure. Conversely, we study the barycentric projection of a plan γ, defined as

ℬ(γ)(x_{1}) := ∫ x_{2}d(γ)_{x_{1}},

showing that it is a well defined map from 𝒫_{2}(ℝ^{2d})_{μ} onto L_{μ}^{2}, and that the range of Tan_{μ}(𝒫_{2}(ℝ^{d})) is Tan_{μ}(𝒫_{2}(ℝ^{d})), thus providing a natural left inverse to ℑ.

The fifth section is devoted to the study of the algebraic properties of operations on 𝒫_{2}(ℝ^{2d})_{μ}: we introduce a (multivalued) sum between plans as

γ_{1} ⊕ γ_{2} := {(π^{0}, π^{1} + π^{2})_{#}α : α ∈ 𝒫_{2}(ℝ^{3d}), π_{#}^{0,1}α = γ_{1}, π_{#}^{0,2}α = γ_{2}},

the product of a plan γ with a scalar λ as

λ · γ := (π^{1}, λπ^{2})_{#}γ,

and we show that these operations and the scalar product defined in (0.14), endow 𝒫_{2}(ℝ^{2d})_{μ} with a structure very similar to the one of an Hilbert space. A remarkable result of this section is that the opposite −1 · γ of a tangent plan γ, is tangent, too: for this reason we called Tan_{μ}(𝒫_{2}(ℝ^{d})) tangent space rather than tangent cone.

The last section of the Chapter studies the projection operator

(0.15)  𝒫_{2}(ℝ^{2d})_{μ} ∋ γ → P(γ) ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})),

where P(γ) is the element of minimal distance from γ among those in Tan_{μ}(𝒫_{2}(ℝ^{d})). We show that this operator is well-defined, that is characterized by the property

⟨γ, η⟩ = ⟨P(γ), η⟩, ∀η ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})),

and that, at least for plans γ induced by maps, it holds the identity

lim_{t→0^{+}} \frac{W (exp(t · γ), μ)}{t} = ∥P(γ)∥.

In Chapter 5 we study the τ-closure properties of geodesically convex sets and of strongly geodesically convex sets in 𝒫_{2}(ℝ^{d}), i.e. those sets C such that

• ((1 − t)π^{1} + tπ^{2})_{#}γ ∈ C for every μ, ν ∈ C and some γ ∈ Opt(μ, ν),
• ((1 − t)π^{1} + tπ^{2})_{#}γ ∈ C for every μ, ν ∈ C and every γ ∈ Adm(μ, ν),

respectively. The reason of such an interest comes from the fact that, as pointed out in [11], in order to have convergence of the minimizing movements scheme it is necessary to have some sort of weak compactness of the sublevels of the functional. On Hilbert spaces this is guaranteed when the functional is convex and lower semicontinuous, because closed and convex sets are weakly closed, thus it is natural to ask whether the same is true for geodesically convex functionals, i.e. functionals convex along constant speed geodesics, in 𝒫_{2}(ℝ^{d}). In [11] mainly two kind of geodesically convex functionals on 𝒫_{2}(ℝ^{d}) are studied: those geodesically convex and those strongly geodesically convex, defined respectively as those functionals F satisfying

• F (μ_{t}) ≤ (1 − t)F (μ_{0}) + tF (μ_{1}) for every μ_{t} = ((1 − t)π^{1} + tπ^{2})_{#}γ, for some optimal plan γ,

• F (μ_{t}) ≤ (1 − t)F (μ_{0}) + tF (μ_{1}) for every μ_{t} = ((1 − t)π^{1} + tπ^{2})_{#}γ, for some plan γ ∈ Adm(μ, ν).

Clearly the sublevels of a (strongly) geodesically convex functional are (strongly) geodesically convex sets, thus in order to have the desired convergence of the minimizing movements technique one should prove some kind of τ−compactness of bounded (strongly) geodesically convex sets. In [11] the problem is solved in two different ways for strongly and not strongly geodesically convex functionals:

• for strongly g.c. functionals an a priori estimate of the distance between two discrete solutions is given, proving the strong convergence of scheme without any compactness assumption,

• for g.c. functionals the τ−compactness of sublevel sets is assumed as a further hypothesis.

The approach we present here is rather different: we focus on (strongly) g.c. sets and we imitate the proof of weak compactness of convex sets in Hilbert spaces to our space, to gain τ−closure. This requires first the introduction of the “halfspaces” ℋ_{γ;C}^{+} and ℋ_{γ;C}^{−}:

ℋ_{γ;C}^{+} := {ν ∈ 𝒫_{2}(ℝ^{d}) : ⟨η, γ⟩ ≥ C, for some η ∈ exp_{μ}^{−1}(ν)} ,
ℋ_{γ;C}^{−} := {ν ∈ 𝒫_{2}(ℝ^{d}) : ⟨η, γ⟩ ≤ C, for some η ∈ exp_{μ}^{−1}(ν)} .

We prove the τ−compactness of the sets ℋ_{γ;C}^{+} and, when the base measure μ is regular, of ℋ_{γ;C}^{−}. And finally we prove that

• a strongly g.c. set may be written as the intersection of a family of hyperplanes of the kind ℋ_{γ;C}^{+} ,

• a g.c. set included in the set of regular measures may be written as the intersection of a family of hyperplanes of the kind ℋ_{γ;C}^{−}.

This shows that strongly g.c. sets are always τ−closed (consistently with what proved in [11]), and that g.c. sets made of regular measures are τ−closed, too: this implies the redundancy of the hypothesis on τ−compactness of the sublevels, at least for functionals which are equal to +∞ on non-regular measures. Technical difficulties arise when trying to generalize the second result to general g.c. sets: it is not clear to the author whether the regularity hypothesis on the measures may be dropped or not.

In the final section of the Chapter we introduce the concept of subdifferential of functionals in 𝒫_{2}(ℝ^{d}) (already presented in [11]), in light of the results and of the notation developed in Chapter 4: again, here we do not introduce new concepts, but just review them in a (possibly) clearer way. The purpose is to show that of the two potential definitions of subdifferential of geodetically convex functional

γ ∈ ∂^{−}F (μ) ⇔ ∀ν ∈ 𝒫_{2}(ℝ^{d}) ∀η ∈ exp_{μ}^{−1}(ν) F (ν) ≥ F (μ) + ⟨γ, η⟩,
γ ∈ ∂^{−}F (μ) ⇔ ∀ν ∈ 𝒫_{2}(ℝ^{d}) ∀η ∈ exp_{μ}^{−1}(ν) F (ν) ≥ F (μ) − ⟨ − 1 · γ, η⟩,

only the second one (equivalent to the one adopted in [11]) is correct, if one aims to have a strong-weak closure property of the subdifferential: this is proved by an explicit counterexample. At the end of the section we apply the definitions given to the study of the differential properties of the functional W^{2}(·, σ).

In Chapter 6 we make a second-order analysis of (𝒫_{2}(ℝ^{d}), W). Given that we know that (𝒫_{2}(ℝ^{d}), W) has a “Riemannian like” structure, we try to answer to questions as: is there a Levi-Civita connection in this case? If yes, what is it? Does parallel transport of vectors exist? What about the curvature operator? These questions, mostly at a formal (smooth) level and on compact Riemannian manifolds, have been already addressed in [43]: here we focus also on some analytic aspects of these questions.

The Chapter is structured as follows: in the first section we recall the case of a regular manifold M embedded in ℝ^{N}, showing how the Levi-Civita connection, the parallel transport and the curvature operator look like in this well-known case. We also show an existence proof of the parallel transport based on an approximation argument, which we will be able to imitate in the Wasserstein setting. The idea is to show that, given a curve [0, 1] ∋ t → γ(t) ∈ M, a tangent vector u_{0} in γ(0), and letting P_{t} be the projection of ℝ^{N} onto the tangent space at M in γ(t), there exists the limit of

P(u_{0}) := P_{1}(P_{t_{n−i}}(⋯ P_{t_{1}}(u_{0})))

over the directed set of partitions P = {0 ≤ t_{0} < ⋯ < t_{n} = 1} of [0, 1] (where with some abuse of notation we wrote P both for the partition and for the induced operator) and that this limit is the parallel transport of u_{0} to the tangent space in γ(1) along the curve γ(t). The key ingredient to prove this result is the inequality

(0.18)  |P_{t_{3}}(u) − P_{t_{3}}(P_{t_{2}}(u))| ≤ C|u||t_{3} − t_{2}||t_{2} − t_{1}|,

valid for all vectors u in the tangent space of γ(t_{1}), where C is a constant which depends on the regularity of the manifold and of the curve. The idea to prove this inequality is that in a regular setting the norm of the operators

V_{t} ∋ u → u − P_{s}(u),
V_{t}^{⊥} ∋ u → P_{s}(u),

are bounded by C|s − t|, where V_{t}, V_{t}^{⊥} are the tangent and the orthogonal space to M at γ(t), respectively. From inequality (0.18), it follows that for given a partition P = {0 ≤ t_{0} ≤ ⋯ ≤ t_{n} = 1} and any refinement P′ it holds

|P(u) − P′(u)| ≤ C|u| ∑_{i=0}^{n−1} |t_{i+1} − t_{i}|^{2},

from which the desired convergence of the approximation scheme follows.

In the second section we develop the analytical tools needed to replicate in the Wasserstein setting the construction made for embedded Riemannian manifolds. The key concept we introduce is the angle between tangent spaces through a map: for a given couple μ, ν ∈ 𝒫_{2}(ℝ^{d}) and a map T ∈ L_{μ}^{2} such that T_{#}μ = ν, we define the angle θ_{T} (μ, ν) ∈ [0, π/2] as

sin θ_{T} (μ, ν) := sup_{{v∈Tanν (𝒫_{2}(ℝ^{d})) \atop ∥v∥ν =1}} ∥v ∘ T − P_{μ}(v ∘ T)∥_{μ},

and we prove that it holds:

sin θ_{T} (μ, ν) ≤ Lip(T − Id).

In the third section we introduce the class of regular curves which is the class of curves along which we are able to define the parallel transport and to show its existence. We will say that an absolutely continuous curve [0, 1] ∋ t → μ_{t} is regular if its velocity vector field v_{t} satisfies:

∫_{0}^{1} Lip(v_{t})dt < ∞,

where by Lip(v), v ∈ L_{μ}^{2}, we mean the infimum of the Lipschitz constant among all the vector field μ−a.e. equal to v.

The well-known Cauchy-Lipschitz theory ensures that for a regular curve there exists a unique family of maps T(t, s, ·) : supp μ_{t} → supp μ_{s}, which we call the flow of the curve, absolutely continuous in s and Lipschitz in x satisfying

(0.20)  \{ \begin{matrix}T(t, t, x) = x, & ∀x, t, \\ \frac{d}{ds}T(t, s, x)|_{s=\overline{s}} = v\overline{s}(T(t, \overline{s}, x)) & ∀x, t, a.e. \overline{s}, \\ T(s, r, T(t, s, x)) = T(t, r, x), & ∀x, t, s, r \\ (T(t, s, x))_{#}μ_{t} = μ_{s} & ∀t, s ∈ [0, 1]. \end{matrix}

The existence of the flow maps allows the introduction of the concept of absolutely continuous vector field along μ_{t}: we say that t → u_{t} ∈ L_{μ_{t}}^{2} is absolutely continuous if for any t_{0} ∈ [0, 1] the map t → u_{t}∘T(t_{0}, t, ·) ∈ L_{μ_{t_{0}}}^{2} is absolutely continuous. Given that the composition with T(t_{0}, t′_{0}, ·) is an isometry between L_{μ_{t′_{0}}}^{2} and L_{μ_{t_{0}}}^{2} , to check the absolute continuity of a vector field, it is sufficient to check whether t → u_{t} ∘ T(t_{0}, t, ·) ∈ L_{μ_{t_{0}}}^{2} is absolutely continuous for some t_{0} ∈ [0, 1].

The derivative \frac{d}{dt}u_{t} ∈ L_{μ_{t}}^{2} of an absolutely continuous vector field u_{t} is then defined as

\frac{d}{dt}u_{t} := lim_{h→0} \frac{u_{t+h} ∘ T(t, t + h, ·) − u_{t}}{h} = \frac{d}{dr} (u_{r} ∘ T(s, r, ·))|_{r=t} ∘ T(t, s, ·), ∀s ∈ [0, 1].

It is a simple consequence of the definition that the derivative exists for a.e. t ∈ [0, 1] and that the function t → ∥\frac{d}{dt}u_{t}∥_{μ_{t}} is integrable. Another easy consequence of the definition is that the Leibnitz rule holds in the following sense:

(0.21)  \frac{d}{dt}⟨u_{t}^{1}, u_{t}^{2}⟩_{μ_{t}} = ⟨\frac{d}{dt}u_{t}^{1}, u_{t}^{2}⟩_{μ_{t}} + ⟨u_{t}^{1}, \frac{d}{dt}u_{t}^{2}⟩_{μ_{t}},

It is important to underline that this definition of derivative allows us to take derivative of a function u_{t} whose range belongs to different L^{2} spaces as t varies: actually these spaces can be quite different from each other, if the support of μ_{t} depends on time.

We conclude the section with a proof of density of regular curves in the set of absolutely continuous curves: the delicate point in our approximation result is due to the fact that regularity imposes a Lipschitz condition on the tangent velocity field. The typical approximation schemes for solutions to the continuity equation, on the other hand, produce a regularized vector field that is compatible with the regularized density, but it is not tangent in general. Therefore a further projection of the regularized velocity on the tangent space is needed.

In section 4 we study the problem of the parallel transport: we prove that along regular curves a precise notion of parallel transport can be given, and we are able to prove its existence and uniqueness. Conversely, we show that under some circumstances the parallel transport “no matter the definition” doesn’t exist if the curve fails to be regular.

Imitating the definition of parallel transport in the case of an embedded manifold, we say that t → u_{t} is a parallel transport along the regular curve μ_{t} if it is tangent, absolutely continuous and satisfies:

(0.22)  P_{μ_{t}} (\frac{d}{dt}u_{t}) = 0, a.e. t ∈ [0, 1],

where P_{μ_{t}} : L_{μ_{t}}^{2} → Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})) is the orthogonal projection. In the case when μ_{t} = ρ_{t}ℒ^{d} (ℒ^{d} being the Lebesgue measure), the PDE corresponding to the parallel transport of a gradient vector field ∇φ_{t} is, in accordance with the formal calculations made in [43],

∇ · ((∂_{t}∇φ_{t} + ∇^{2}φ_{t} · v_{t})ρ_{t}) = 0.

It is a simple consequence of the definition and of equation (0.21) that the parallel transport is linear and preserves the scalar product, and therefore is unique.

Existence is a more delicate issue. The key bound we use, which is a direct consequence of equations (0.20) is

(0.23)  Lip(T(t, s, ·) − Id) ≤ exp (∫_{t}^{s} Lip(v_{r})dr) − 1.

The idea of the proof is to imitate the construction made for embedded manifolds: we introduce the operators 𝒫_{t}^{s} : Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})) → Tan_{μ_{s}}(𝒫_{2}(ℝ^{d})) as 𝒫_{t}^{s}(u) := P_{μ_{s}}(u ∘ T(s, t, ·)), then we observe that for any u ∈ Tan_{μ_{0}}(𝒫_{2}(ℝ^{d})) the vector field t → 𝒫_{0}^{t}(u) is a first order approximation of the parallel transport, in the sense that it satisfies equation (0.22) at t = 0. Indeed from the bound (0.23) it follows

∥𝒫_{t}^{0}(𝒫_{0}^{t}(u)) − u∥_{μ_{0}} = ∥𝒫_{t}^{0}(𝒫_{0}^{t}(u) − u ∘ T(t, 0, ·))∥_{μ_{0}} ≤ ∥u∥_{μ_{0}} (∫_{0}^{t} Lip(v_{r})dr)^{2} = o(t).

Then we define for any partition P := {0 = t_{1}, …, t_{n} = 1} of [0, 1] the map from Tan_{μ_{0}}(𝒫_{2}(ℝ^{d})) to Tan_{μ_{1}}(𝒫_{2}(ℝ^{d}))

P(u) := 𝒫_{t_{n−1}}^{t_{n}}(⋯ 𝒫_{t_{1}}^{t_{2}}(u)),

and, thanks to the bound (0.23), we are able to prove that for any u ∈ Tan_{μ_{0}}(𝒫_{2}(ℝ^{d})) there exists the limit T_{0}^{1}(u) of P(u) as P varies over the direct set of partitions. We then repeat this argument to the restriction of the curve to the intervals [t, s], to produce a family of maps T_{t}^{s} : Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})) → Tan_{μ_{s}}(𝒫_{2}(ℝ^{d})). These maps are exactly the parallel transport maps along the regular curve μ_{t}, in the sense that for any u ∈ Tan_{μ_{0}}(𝒫_{2}(ℝ^{d})) the vector field t → T_{0}^{t}(u) is the parallel transport of u along μ_{t}.

We conclude the section with some explicit example of parallel transport and with a counterexample to its existence along a non regular curve.

In the fifth section we define the covariant derivative and the curvature tensor on the space (𝒫_{2}(ℝ^{d}), W). The spirit is similar to the one of Lott in [43], although with a less formal approach.

Recalling that in a classical embedded Riemannian manifold the parallel transport T_{t}^{s} : T_{γ(t)}M → T_{γ(s)}M and the covariant derivative ∇_{γ̇(t)} along a curve γ(t) are related by

∇_{γ̇(t)}u_{t} = lim_{s→t} P_{t} (\frac{T_{s}^{t}(u_{s}) − u_{t}}{s − t} ) ,

we define analogously the covariant derivative in 𝒫_{2}(ℝ^{d}) along the regular curve μ_{t} as

∇_{v_{t}}u_{t} = lim_{s→t} P_{μ_{t}} (\frac{T_{s}^{t}(u_{s}) − u_{t}}{s − t} ) = P_{μ_{t}}(\frac{d}{dt}u_{t}),

for any absolutely continuous vector field t → u_{t} (here v_{t} is the velocity vector field of the curve μ_{t}). We are then able to prove that this derivative is the Levi-Civita on 𝒫_{2}(ℝ^{d}), proving that it is uniquely characterized by the compatibility with the metric and the torsion-free identity.

Having defined the Levi-Civita connection, we can study the curvature operator: we will give a useful representation formula for it (already appeared in [43]) and prove that the sectional curvatures of 𝒫_{2}(ℝ^{d}) are always non-negative.

In section 6 we continue analyzing the formal analogy between 𝒫_{2}(ℝ^{d}) and an embedded Riemannian manifold, by defining an analog of the Sasaki metric on the tangent bundle.

Recall that, for a Riemannian manifold M, it is possible to endow the tangent bundle T M with a natural Riemannian metric, the so-called Sasaki metric, which induces the following distance on the tangent bundle:

d^{2}((p^{1}, u^{1}), (p^{2}, u^{2})) = inf_{γ} {(L(γ))^{2} + |T (u^{1}) − u^{2}|^{2}} ,

where the infimum is taken among all the smooth curves γ(t) in M connecting p^{1} to p^{2}, L(γ) is the length of γ and T (u^{1}) is the parallel transport of u^{1} along γ to the point p^{2}.

This definition has an analogous in (𝒫_{2}(ℝ^{d}), W_{2}), because it doesn’t rely on a differential structure on the tangent bundle to be written. Therefore, for any couple of measures μ, ν ∈ 𝒫_{2}(ℝ^{d}) and any couple of tangent vectors u ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})), v ∈ Tan_{ν}(𝒫_{2}(ℝ^{d})) we define

d^{2}((μ, u), (ν, v)) := inf_{μ_{t}} {(L(μ_{t}))^{2} + ∥v − T_{0}^{1}(u)∥_{ν}^{2}},

where the infimum is taken on the set of regular curves μ_{t} : [0, 1] → 𝒫_{2}(ℝ^{d}) such that μ_{0} = μ and μ_{1} = ν, L(μ_{t}) is the length of μ_{t} and T_{t}^{s} are the parallel transport maps along μ_{t}. In particular we define d((μ, u), (ν, v)) := +∞ if there is no regular curve connecting μ to ν (which may happen, think for instance to the case in which the the support of μ and the one of ν have a different number of connected components: the fact that the flow maps are isomorphisms implies that there is no regular curve joining μ to ν).

In order to obtain a real valued distance we first define the lower semicontinuous envelope d^{∗} of d by:

d^{∗}((μ, u), (ν, v)) := inf{lim inf_{n→∞} d((μ_{n}, u_{n}), (ν_{n}, v_{n})) :

(μ_{n}, u_{n}) → (μ, u), (ν_{n}, v_{n}) → (ν, v)},

where by convergence of (μ_{n}, u_{n}) to (μ, u) we mean: W (μ, μ_{n}) → 0, ∫ u_{n}φdμ_{n} → ∫ uφdμ, ∀φ ∈ C^{b}(ℝ^{d}, ℝ^{d}) and ∥u_{n}∥_{μ_{n}} → ∥u∥_{μ}. Then, since it is not clear whether the function d^{∗} is sufficienly well-behaved (for instance whether the triangle inequality holds), we define the function D as:

D((μ, u), (ν, v)) := inf {d^{∗}((μ, ∇φ), (ν, ∇ψ)) + ∥u − ∇φ∥_{μ} + ∥v − ∇ψ∥_{ν}} ,

where the infimum is taken among all φ, ψ ∈ C_{c}^{∞}(ℝ^{d}). We then prove that D is a distance on the set of couples (μ, u), u ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})), which metrizes the convergence described above. Furthermore, for any absolutely continuous curve (μ_{t}, u_{t}) w.r.t. D the curve μ_{t} is absolutely continuous in 𝒫_{2}(ℝ^{d}) and:

(0.24)  lim_{s→t} \frac{D((μ_{s}, u_{s}), (μ_{t}, u_{t}))}{|s − t|} ≥ ∥v_{t}∥_{μ_{t}}  for a.e. t,

where v_{t} is the tangent field of μ_{t}. Conversely, if μ_{t} is a regular curve and u_{t} is a parallel transport along it, the map (μ_{t}, u_{t}) is absolutely continuous w.r.t. D and equality holds a.e. in (0.24).

Having this result in mind, we give the following definition: t → u_{t} is a weak parallel transport along the absolutely continuous curve t → μ_{t} if the curve t → (μ_{t}, u_{t}) is absolutely continuous w.r.t. D and equality holds in (0.24) for a.e. t. The final part of the section is devoted to the study of the properties of the weak parallel transport. However, it is important to underline that even this weaker notion of transport is not sufficient to gain existence: the counterexample to the existence of parallel transport we present in section 4, works as well for this weaker notion.

CHAPTER 1 Preliminary results

1. Transport of measures

For any separable metric space X we will denote by ℬ(X) the family of all Borel subset of X, and by 𝒫(X) the set of Borel probability measures on X. Given two separable metric spaces X_{1}, X_{2}, a measure μ ∈ 𝒫(X_{1}) and a μ−measurable map T : X_{1} → X_{2} the push forward of μ through T is the measure T_{#}μ ∈ 𝒫(X_{2}) defined as

(1.1)  T_{#}μ(E) := μ(T^{−1}(E)) ∀E ∈ ℬ(X_{2}).

Note that if f, g are T_{#}μ-measurable functions which differ on a T_{#}μ-negligible set E then f ∘ T, g ∘ T differ on T^{−1}(E) which is μ-negligible by definition of T_{#}μ. In particular, for every T_{#}μ-measurable function f, the function f ∘ T is μ−measurable, and it follows from the definition that

(1.2)  ∫_{X_{2}} fdT_{#}μ = ∫_{X_{1}} f ∘ T dμ.

More precisely, one integral exists if and only if exists the other, and in this case they are equal. Observe that for any p ≥ 1 the composition with T is a well defined map from ℒ^{p}(X_{2}, T_{#}μ) to ℒ^{p}(X_{1}, μ): equation (1.2) implies that this map is an isometry. The same argument shows that if S : X_{2} → X_{3} is a T_{#}μ measurable map with values in a separable metric space X_{3}, then T ∘ S is a well defined μ−measurable map. In this case it holds

(S ∘ T)_{#}μ = S_{#}(T_{#}μ).

From now on we will denote with supp μ the support of the measure μ ∈ 𝒫(X_{1}) defined as the complement of the largest open set U such that μ(U) = 0, or equivalently, as

supp μ := {x ∈ X_{1} : μ(U) > 0 for each neighborhood U of x}.

It is always true that the measure T_{#}μ is concentrated on the image of T, however it may happen that supp(T_{#}μ) ⊊ \overline{T (supp μ)} if the image of T is not closed. A simpler situation occurs if we let T be continuous: indeed in this case it holds

T (supp μ) ⊂ supp(T_{#}μ) = \overline{T (supp μ)}.

Moreover it is easy to check that under this hypothesis the map T_{#} : 𝒫(X_{1}) → 𝒫(X_{2}) is continuous w.r.t. the weak convergence in duality with continuous and bounded functions, that is:

μ_{n} ⇀ μ ⇔ ∫_{X} φdμ_{n} → ∫_{X} φdμ, ∀φ ∈ C_{b}(X).

In the following we will often consider product spaces X = ∏_{1}^{N} X_{i} where the X_{i} are separable metric spaces (mostly copies of ℝ^{d}), in this case we will usually write x_{i} and x = (x_{1}, …, x_{N}) for a variable in X_{i} and X respectively. The projections operators π^{i} and π^{i_{1},...,i_{k}} are defined as

\begin{aligned}π^{i} : X & → & X_{i} & π^{i_{1},...,i_{k}} : X & → & ∏_{j=1}^{k}X_{i_{j}} \\ x & → & x_{i} & x & → & (x_{i_{1}}, …, x_{i_{k}}). \end{aligned}

Given a measure μ ∈ 𝒫(X) (usually called N−plan), its marginals are defined as π_{#}^{i}μ ∈ 𝒫(X_{i}) and π_{#}^{i_{1},...,i_{k}}μ ∈ 𝒫(∏_{k} X_{i_{k}}). For 𝒜_{i} ⊂ 𝒫(X_{i}), i = 1, …, N, the set of admissible N−plans is

ADM(𝒜_{1}, …, 𝒜_{N}) := {μ ∈ 𝒫(X) : π_{#}^{i}μ ∈ 𝒜_{i}, ∀i = 1, … N}.

If each 𝒜_{i} is a singleton, 𝒜_{i} = {μ_{i}}, then we will write ADM(μ_{1}, …, μ_{N}) instead of ADM({μ_{1}}, …, {μ_{N}}).

Remark 1.1. If (μ_{n}) ∈ 𝒫(X) is a sequence of plans weakly converging in duality with C_{b} to some μ ∈ 𝒫(X) then the continuity of the projection implies that π_{#}^{i}μ_{n} weakly converge in duality with C_{b} to π_{#}^{i}μ for any i = 1, …, N.

If N = 2 then a plan μ ∈ ADM(μ_{1}, μ_{2}) is also called a transport plan for the couple (μ_{1}, μ_{2}). If a transport plan satisfies

(1.3)  μ = (Id, T)_{#}μ_{1}

for some map T, then we will say that the plan is induced by the map T from μ_{1}. It can be easily seen that a plan μ is induced by a map if and only if it is concentrated on the graph of some μ_{1}−measurable map T, moreover in this case the map T is unique up to μ_{1}−negligible sets and (1.3) holds.

2. Tightness, 2-boundedness and 2-uniform integrability

From now on we will work in Euclidean spaces, so we assume X = ℝ^{d}. A subset 𝒜 ⊂ 𝒫(X) is said to be tight if for any ε > 0 there exists a compact subset K_{ε} of X such that μ(X \ K_{ε}) < ε for every μ ∈ 𝒜, or, equivalently, if

\overline{lim}_{R→+∞} sup_{μ∈𝒜} μ(X \ B_{R}) = 0.

A different, but equivalent, formulation of the definition of tightness is the following: 𝒜 is tight if and only if there exists a positive function φ such that φ(x) → +∞ as |x| → +∞ for which it holds

(1.4)  sup_{μ∈𝒜} ∫ φdμ =: C < +∞.

Indeed suppose (1.4) holds, then Chebichev inequality gives

μ(\{φ > t\}) ≤ \frac{C}{t} ,

and since the set {φ ≤ t} is compact we get one implication. Conversely suppose that 𝒜 is tight: let K_{n} be an increasing sequence of compact sets such that μ(X \K_{n}) ≤ 2^{−n} for any μ ∈ 𝒜 and define

φ(x) := inf {n : x ∈ K_{n}}.

Clearly φ → +∞ tends to infinity as |x| → ∞, moreover for any μ ∈ 𝒜 it holds

∫ φdμ = ∑_{n} n(μ(K_{n} \ K_{n−1})) = ∑_{n} n(μ(X \ K_{n−1}) − μ(X \ K_{n})) ≤ ∑_{n} \frac{n}{2^{n−1}} = 1,

where K_{0} = ∅.

Theorem 1.2 (Prokhorov). A set 𝒜 ⊂ 𝒫(X) is relatively compact w.r.t. the weak convergence in duality with C_{b} if and only if it is tight. Moreover a sequence (μ_{n}) ∈ 𝒫(X) weakly converging in duality with C_{c} to a measure μ (possibly not in 𝒫(X)) is tight if and only if

(1.5)  lim_{n} ∫ φdμ_{n} = ∫ φdμ, ∀φ ∈ C_{b}(X).

Proof. Suppose that 𝒜 is tight, choose a sequence μ_{n} ∈ 𝒜 and extract a subsequence (not relabeled) weakly converging in duality with C_{c} to some μ. Clearly μ(X) ≤ 1, so we need to prove that μ(X) ≥ 1. Fix ε > 0, find a compact K_{ε} such that μ_{n}(K_{ε}) ≥ 1−ε for any n, and choose a function χ_{ε} ∈ C_{c}(X) such that 0 ≤ χ_{ε} ≤ 1 and identically 1 on K_{ε}. Since it holds

(1.6)  μ(X) ≥ ∫ χ_{ε}dμ = lim_{n} ∫ χ_{ε}dμ_{n} ≥ \underline{lim} μ_{n}(K_{ε}) ≥ 1 − ε,

we proved that tightness implies compactness. Let us turn to the converse implication: suppose 𝒜 is not tight so that there exists ε > 0 such that for any compact set K it holds sup_{n} μ_{n}(X \ K) > ε. Up to pass to a subsequence, not relabeled, we may assume that (μ_{n}) weakly converge to some μ in duality with C_{c}. Choose R_{n} ↗ +∞ and, for any n, find a measure μ_{n} ∈ 𝒜 such that μ_{n}(B_{R_{n}}) ≤ 1 − ε. Letting χ_{n} ∈ C_{c}(X) be a function such that 0 ≤ χ_{n} ≤ 1, χ ≡ 1 on B_{R_{n}} and χ_{n} ≡ 0 outside B_{R_{n+1}} we have

μ(X) = sup_{n} μ(B_{R_{n}}) ≤ sup_{n} ∫ χ_{n}dμ = sup_{n} lim_{m} ∫ χ_{n}dμ_{m} ≤ 1 − ε.

For the second part of the proof note that choosing φ ≡ 1 in (1.5) we get μ(X) = 1 and so, by the previous argument, that μ_{n} is tight. Suppose conversely that μ_{n} is tight, fix φ ∈ C_{b}(X) and let χ_{R} ∈ C_{c}(x) be a cut-off function 0 ≤ χ_{R} ≤ 1 identically 1 in B_{R} and equal to 0 outside B_{2R}. We have

\begin{matrix}|∫ φdμ_{n}−∫ φdμ| ≤ & |∫ φχ_{R}dμ_{n}−∫ φχ_{R}dμ| + |∫ φ(1 − χ_{R})dμ_{n}| + |∫ φ(1 − χ_{R})dμ| \\ ≤ & |∫ φχ_{R}dμ_{n}−∫ φχ_{R}dμ| + sup |φ|( sup_{n} μ_{n}(X \ B_{R}) + μ(X \ B_{R})), \end{matrix}

from which it follows

\overline{lim}_{n} |∫ φdμ_{n} − ∫ φdμ| ≤ sup |φ|( sup_{n} μ_{n}(X \ B_{R}) + μ(X \ B_{R})).

Letting R tend to +∞ we get the thesis. □

Proposition 1.3. Let X = ∏_{i=1}^{N} X_{i} be a product space. A set 𝒜 ⊂ 𝒫(X) is tight if and only if so are the sets π_{#}^{i}(𝒜) ⊂ 𝒫(X_{i}) for any i = 1, …, n.

Proof. One implication follows from the relative compactness of 𝒜 and the continuity of the maps π_{#}^{i} w.r.t. the duality with C_{b}. For the other one fix ε > 0 and choose compact sets K_{i} ⊂ X_{i} such that μ_{i}(X_{i} \ K_{i}) ≤ ε/N for any μ_{i} ∈ π_{#}^{i}(𝒜), i = 1, …, n. The set ∏_{i} K_{i} is compact in X and it holds

μ(X \ ∏_{i=1}^{n} K_{i}) ≤ ∑_{i=1}^{n} μ((π^{i})^{−1}(X \ K_{i})) ≤ ε, ∀μ ∈ 𝒜 .

□

In the next chapters we will mostly work with the subset of 𝒫(X) of measures with finite second moment:

𝒫_{2}(X) := {μ ∈ 𝒫(X) : ∫ |x|^{2}dμ < +∞}.

Note that if μ ∈ 𝒫_{2}(X) and φ is a Borel function with quadratic growth, i.e. a function such that

|φ(x)| ≤ A|x|^{2} + B ∀x ∈ X,

for some A ≥ 0, B ∈ ℝ, then φ ∈ ℒ^{1}(X, μ) and so ∫ φdμ is a well defined real number.

A set 𝒜 ⊂ 𝒫_{2}(X) is 2-uniformly integrable if for every ε there exists a compact set K_{ε} such that

∫_{X\K_{ε}} |x|^{2}dμ < ε, ∀μ ∈ 𝒜 .

This definition is formally very close to the one of tightness (it is actually possible to introduce the 2-uniform integrability in terms of tightness of the set {(1 + |x|^{2})μ : μ ∈ 𝒜 }), and it will be of primary importance in the study of compactness in the space 𝒫_{2}(X) endowed with the Wasserstein distance.

The proof of the next proposition is very similar to the one we gave before, so we will omit it.

Proposition 1.4. Given 𝒜 ⊂ 𝒫_{2}(X) the following three conditions are equivalent. i) 𝒜 is 2-uniformly integrable. ii)

\overline{lim}_{R→+∞} sup_{μ∈𝒜} ∫_{X\B_{R}} |x|^{2}dμ(x) = 0.

iii)

sup_{μ∈𝒜} ∫ φdμ < +∞,

for some positive function φ with more than quadratic growth at infinity, i.e. a function such that φ(x)/|x|^{2} → +∞ as |x| → +∞.

Moreover for a given sequence (μ_{n}) ⊂ 𝒫_{2}(X) weakly converging in duality with C_{c} to some μ ∈ 𝒫_{2}(X), the following conditions are equivalent: i) (μ_{n}) is 2-uniformly integrable,

ii) ∫ |x|^{2}dμ = lim ∫ |x|^{2}dμ_{n},

iii) ∫ φdμ = lim ∫ φdμ_{n} for any continuous function φ with quadratic growth at

infinity.

A set 𝒜 ⊂ 𝒫_{2}(X) is 2-bounded if

sup_{μ∈𝒜} ∫ |x|^{2}dμ(x) < +∞.

Obviously a 2-uniformly integrable set is 2-bounded. It is easy to prove that a set is 2-bounded if and only if

sup_{μ∈𝒜} ∫ φdμ < +∞,

for any φ with quadratic growth at infinity. Moreover for a 2-bounded sequence (μ_{n}) ⊂ 𝒫_{2}(X) weakly converging to μ ∈ 𝒫(X) it holds

(1.7)  ∫ φdμ = lim_{n} ∫ φdμ_{n},

for any continuous function φ with less than quadratic growth, i.e. a function such that φ(x)/(1 + |x|^{2}) → 0 as |x| → +∞.

Proposition 1.5. Let 𝒜 ⊂ 𝒫_{2}(X) be a set of probability measures on a product space X = ∏ X_{i}, i = 1, …, N. Then 𝒜 is 2-uniformly integrable (2-bounded) if and only if so are the sets π_{#}^{i}(𝒜) ⊂ X_{i}, i = 1, …, N.

Proof. The assertions about 2-boundedness follow from

∫_{X} ∑_{i=1}^{N} |x_{i}|^{2}dμ = ∑_{i=1}^{N} ∫_{X_{i}} |x_{i}|^{2}dπ_{#}^{i}μ,

so we turn to the 2-uniform integrability. Since

∫_{X\∏ K_{i}} ∑_{i=1}^{N} |x_{i}|^{2}dμ ≥ ∑_{i=1}^{N} ∫_{X\(π^{i})^{−1}(K_{i})} |x_{i}|^{2}dμ = ∑_{i=1}^{N} ∫_{X\(π^{i})^{−1}(K_{i})} |x_{i}|^{2}dπ_{#}^{i}μ

for any i = 1, …, N, the 2-uniform integrability of 𝒜 implies the one of π_{#}^{i}(𝒜) for any i = 1, …, N. For the other implication, by induction we can assume N = 2. Fix R, let B_{R}^{i} ⊂ X_{i}, i = 1, 2, be the balls with center in the origin and radius R and use the set equality

X \ (B_{R}^{1} × B_{R}^{2}) =

((X_{1} \ B_{R}^{1}) × B_{R}^{2}) ⋃ (B_{R}^{1} × (X_{2} \ B_{R}^{2})) ⋃ ((X_{1} \ B_{R}^{1}) × (X_{2} \ B_{R}^{2})),

to easily obtain

∫_{X\(B_{R}^{1}×B_{R}^{2})} |x_{1}|^{2} + |x_{2}|^{2}dμ ≤ 3 ∫_{X_{1}\B_{R}^{1}} |x_{1}|^{2}dπ_{#}^{1}μ + 3 ∫_{X_{2}\B_{R}^{2}} |x_{2}|^{2}dπ_{#}^{2}μ,

from which the thesis follows. □

3. Dudley’s lemma

A fundamental tool we will need in the sequel is Dudley’s lemma. In this section we will prove it as a consequence of the disintegration theorem (recall that the spaces we are dealing with are Euclidean). Recall that a map x_{1} ∈ X_{1} → μ_{x} ∈ 𝒫(X_{2}) is called a Borel map if so is the map x_{1} ∈ X_{1} → μ_{x_{1}}(A) for any open set A ⊂ X_{2}. Given such a Borel map and a measure μ ∈ 𝒫(X_{1}), it is easy to see that the formula

I(φ(x_{1}, x_{2})) := ∫ ∫ φ(x_{1}, x_{2})dμ_{x_{1}}(x_{2})dμ(x_{1}),

defines a continuous positive linear functional on C_{c}(X_{1} × X_{2}) with norm 1, which is therefore a probability measure on X_{1} ×X_{2}. It is natural to ask whether this statement has an inverse: the answer is positive and it is given by the following proposition, which is a direct consequence of the disintegration theorem.

Proposition 1.6. Let X_{1}, X_{2} be two Euclidean spaces and μ ∈ 𝒫(X_{1} ×X_{2}). Then there exists a Borel map x_{1} → μ_{x_{1}} on X_{1} with values in 𝒫(X_{2}) such that

∫ φ(x_{1}, x_{2})dμ(x_{1}, x_{2}) = ∫ ∫ φ(x_{1}, x_{2})dμ_{x_{1}}(x_{2})dπ_{#}^{1}μ,

for every φ ∈ C_{c}(X_{1} × X_{2}). Moreover the family μ_{x_{1}} is uniquely determined up to equality π_{#}^{1}μ a.e..

We will refer to the family μ_{x_{1}} as the disintegration of μ with respect to its first marginal (or w.r.t. the first variable) and write dμ(x_{1}, x_{2}) = dπ_{#}^{1}μ(x_{1}) ⊗ dμ_{x_{1}}(x_{2}) or dμ = dπ_{#}^{1}μ ⊗ dμ_{x_{1}} if no ambiguity occurs.

Let us now turn to the main result of this section.

Lemma 1.7 (Dudley). Let X_{1}, X_{2}, X_{3} be three Euclidean spaces and let μ^{1,2} ∈ 𝒫(X_{1} × X_{2}), μ^{1,3} ∈ 𝒫(X_{1} × X_{3}) be two measures such that

(1.8)  π_{#}^{1}μ^{1,2} = π_{#}^{1}μ^{1,3}.

Then there exists a plan μ^{1,2,3} ∈ 𝒫(X_{1} × X_{2} × X_{3}) such that

π_{#}^{1,2}μ^{1,2,3} = μ^{1,2}, (1.9a)

π_{#}^{1,3}μ^{1,2,3} = μ^{1,3}. (1.9b)

Proof. Observe that the condition (1.8) is necessary to the existence of μ^{1,2,3}. To show the sufficency let μ := π_{#}^{1}μ^{1,2} = π_{#}^{1}μ^{1,3} be the common marginal and μ_{x_{1}}^{1,2}, μ_{x_{1}}^{1,3} be the disintegrations of μ^{1,2}, μ^{1,3} w.r.t. μ. It is easy to check that the measure μ^{1,2,3} defined by

(1.10)  dμ^{1,2,3}(x_{1}, x_{2}, x_{3}) := dμ(x_{1}) ⊗ d(μ_{x_{1}}^{1,2}(x_{2}) ⊗ μ_{x_{1}}^{1,3}(x_{3}))

satisfies the condition. □

Observe that if a measure μ^{1,2,3} satisfies (1.9a), (1.9b) then it has to hold

π_{#}^{2}μ_{x_{1}}^{1,2,3} = μ_{x_{1}}^{1,2},
π_{#}^{3}μ_{x_{1}}^{1,2,3} = μ_{x_{1}}^{1,3}.

Together with the uniqueness result stated in 1.6 this fact implies that if either μ^{1,2} or μ^{1,3} is induced by a map on X_{1}, then there is a unique 3-plan satisfying the conditions of 1.7. More explicitely if μ^{1,2} = (Id, T)_{#}μ then the unique μ^{1,2,3} is given by

dμ^{1,2,3}(x_{1}, x_{2}, x_{3}) = dμ(x_{1}) ⊗ d(δ_{T (x_{1})} ⊗ μ_{x_{1}}^{1,3}(x_{3})).

It is usually said that Dudley’s lemma allows to make the composition of plans, the reason being the following. It is just a matter of notation to observe that an analogous version of Dudley lemma holds if the given measures μ^{1,2}, μ^{2,3} belong to 𝒫(X_{1} × X_{2}), 𝒫(X_{2} × X_{3}) respectively and equation (1.8) is replaced by

π_{#}^{2}μ^{1,2} = π_{#}^{2}μ^{2,3}.

In this case if μ^{1,2} = (Id, T)_{#}μ and μ^{2,3} = (Id, S)_{#}(T_{#}μ) for some measure μ ∈ 𝒫(X_{1}) then the unique 3-plan with marginals μ^{1,2}, μ^{2,3} is given by μ^{1,2,3} = (Id, T, (S ∘ T))_{#}μ and it satisfies π_{#}^{1,3}μ^{1,2,3} = (Id, (S ∘ T))_{#}μ, which is exactly the plan induced by the composition of the two given maps.

4. On the solution of the optimal transport problem

In this section we collect the basic results we need to introduce the Wasserstein distance. It is not our purpose to describe accurately how to solve the optimal transport problem neither the techniques used and we do not intend to state the results in their maximum generality. We simply list the main definitions and theorems in the form and setting we are interested in, and we refer to [7],[11], [45], [15] for a more detailed analysis.

We will mostly work with 2-bounded sets, therefore in the following we will not indicate if weak convergence is done w.r.t. C_{b} or C_{c}, exceptions will be indicated.

The optimal transport problem is the following: given two measures μ, ν ∈ 𝒫_{2}(ℝ^{d}) find the infimum of

(1.11)  Adm(μ, ν) ∋ γ → ∫ |x_{1} − x_{2}|^{2}dγ(x_{1}, x_{2}),

and answer to questions like: does there exist a minimizing plan? Is it unique? Is it induced by a map?

A standard tightness-semicontinuity argument ensures that there always exists a plan γ ∈ Adm(μ, ν) which attains the minimum, and simple counterexamples show that in general this plan need not to be unique.

We will denote with Opt(μ, ν) the set of optimal plans for the couple μ, ν:

(1.12) γ ∈ Opt(μ, ν) ⇔

γ ∈ Adm(μ, ν) and ∫ |x_{1} − x_{2}|^{2}dγ = inf_{γ′∈Adm(μ,ν)} ∫ |x_{1} − x_{2}|^{2}dγ′.

Note that by weak lower semicontinuity of γ → ∫ |x_{1} − x_{2}|^{2}dγ it follows that Opt(μ, ν) is weakly closed.

In order to understand the main properties of optimal plans we need the following definition.

Definition 1.8. A set A ⊂ (ℝ_{1}^{d} × ℝ_{2}^{d}) is cyclically monotone if for every N ≥ 1 and any choice of (x_{1}^{i}, x_{2}^{i}) ∈ A, i = 1, …, N, it holds

(1.13)  ∑_{i=1}^{N} ⟨x_{1}^{i}, x_{2}^{i}⟩ ≥ ∑_{i=1}^{N} ⟨x_{1}^{i}, x_{2}^{i+1}⟩,

where x_{2}^{N+1} = x_{2}^{1}.

It can be proved that the following theorem holds .

Theorem 1.9. Let γ ∈ 𝒫_{2}(ℝ_{1}^{d} × ℝ_{2}^{d}) be a plan and μ := π_{#}^{1}γ, ν := π_{#}^{2}γ be its marginals. Then γ ∈ Opt(μ, ν) if and only if its support is cyclically monotone.

Remark 1.10. The following property holds (a little bit stronger then the one stated in the previous theorem): for any μ, ν ∈ 𝒫_{2}(ℝ^{d}) the set

(1.14)  \overline{⋃_{γ∈Opt(μ,ν)} supp γ},

is cyclically monotone. Indeed it is sufficient to consider a weakly dense countable subset {γ_{n}}_{n∈ℕ} of Opt(μ, ν), to consider the (optimal) plans γ′_{N} := 1/N ∑_{1}^{N} γ_{n}, to notice that supp γ′_{N} = ∪_{1}^{N} supp γ_{n} and to pass to the limit to get (1.14).

A simple but of great importance consequence of this theorem is the following stability result.

Proposition 1.11. Let (μ_{n}), (ν_{n}) ⊂ 𝒫_{2}(ℝ^{d}) be two sequences of measures weakly converging to μ, ν respectively and let γ_{n} ∈ Opt(μ_{n}, ν_{n}) be any choice of optimal plans. Then the sequence (γ_{n}) is tight and any weak accumulation point γ belongs to Opt(μ, ν).

Proof. The tightness of the sequence follows from proposition 1.3 and remark 1.1 ensures that any accumulation point γ belongs to Adm(μ, ν). Now suppose that γ_{n} ⇀ γ, observe that under this hypothesis the support of γ is contained in the Kuratowski lower limit of the supports of γ_{n} (i.e. any point of supp γ can be approximated by points in supp γ_{n}): the continuity of the scalar product ensures that supp γ is cyclically monotone, too. □

The following classical result of Rockafellar gives a very interesting characterization of cyclically monotone sets.

Theorem 1.12 (Rockafellar). A subset A ⊂ ℝ^{d} × ℝ^{d} is cyclically monotone if and only if it is contained in the graph of the subdifferential of a convex function.

From this theorem it follows the well known result of Brenier on the existence of optimal transport maps (i.e. maps which induce optimal transport plans). Recall that a set C ⊂ ℝ^{d} is said to be H^{k} rectifiable if there exists a sequence f_{n} of Lipschitz maps from ℝ^{k} to ℝ^{d} such that

H^{k} (C \ ⋃_{n} f_{n}(ℝ^{k})) = 0,

where H^{k} is the k−Hausdorff measure, and the following property of convex functions (see [3]).

Theorem 1.13. Let φ : ℝ^{d} → ℝ be a convex function. Then the set of points of non-differentiabilty is H^{d−1} rectifiable.

Definition 1.14 (Regular measures). The set of regular measures on ℝ^{d} is the subset of 𝒫_{2}(ℝ^{d}) defined as follows

(1.15)  𝒫_{2}^{r}(ℝ^{d}) := {μ ∈ 𝒫_{2}(ℝ^{d}) : μ(C) = 0, for any H^{d−1} rectifiable set C}

Theorem 1.15 (Brenier). Let μ, ν ∈ 𝒫_{2}(ℝ^{d}) be given measures and suppose that μ is regular. Then there exists a unique optimal plan and this plan is induced by the gradient of a convex function.

Proof. Use remark 1.10 and Rockafellar theorem to obtain the existence of a convex function φ such that every optimal plan is concentrated on the graph of ∂^{−}φ (the subdifferential of φ). Now observe that since μ is regular ∂^{−}φ(x) is a singleton for μ-a.e. x, therefore the disintegration of any optimal plan γ is given by dγ(x, y) = dμ(x) × δ_{∇φ(x)}dy. This implies that the optimal plan is unique and it is given by (Id, ∇φ)_{#}μ. □

Remark 1.16 (Discrete measures). Another interesting case in which there is existence of an optimal transport map, although not necessarily unique, is the one in which the two measures μ and ν are both discrete, with finite support of the same cardinality, say N, and each point has mass 1/N. Indeed, it is not hard to see that under these assumptions, the extremal points of the closed convex set of admissible plans are given exactly by those plans induced by maps. From the linearity of the cost, the claim follows.

From now on, whenever μ ∈ 𝒫_{2}^{r}(ℝ^{d}) is a regular measure, we will indicate by T_{μ}^{ν} the optimal transport map from μ to ν ∈ 𝒫_{2}(ℝ^{d}) given by the previous theorem. Observe that if ν is regular too, the uniqueness of the optimal plan ensures

T_{ν}^{μ} ∘ T_{μ}^{ν} = Id, μ − a.e.,
T_{μ}^{ν} ∘ T_{ν}^{μ} = Id, ν − a.e..

In general, however, the group property does not hold:

(1.16)  T_{ν}^{σ} ∘ T_{μ}^{ν} ≠ T_{μ}^{σ},

due to the fact that a composition of monotone maps on ℝ^{d}, d > 1, need not be monotone. As we will see in the next chapter, equation (1.16) is the one which in some sense is responsible of the non-flatness of the manifold 𝒫_{2}(ℝ^{d}), see proposition 2.13 and equations (2.19), (2.20).

CHAPTER 2 The distance W and its geometry

1. The distance W

In this chapter we define the distance W and we investigate the first properties of the space (𝒫_{2}(ℝ^{d}), W) from both topological and geometrical viewpoints.

Definition 2.1. Let μ_{1}, μ_{2} ∈ 𝒫_{2}(ℝ^{d}) be two measures. The Wasserstein distance W (μ_{1}, μ_{2}) between them is defined by

W^{2}(μ_{1}, μ_{2}) := inf_{γ∈Adm(μ_{1},μ_{2})} ∫ |x_{1} − x_{2}|^{2}dγ(x_{1}, x_{2}).

We will call the number \sqrt{∫ |x_{1} − x_{2}|^{2}dγ} the cost of the plan γ.

The discussion of the previous chapter ensures that infimum is always attained: recall that Opt(μ_{1}, μ_{2}) is the set of those plans that reach the minimum.

In order to let the notation be not too heavy, from now on we will indicate by ∥f(x)∥_{γ} the L^{2} norm w.r.t. the measure γ of the function f whose domain contains the support of γ: for instance

∥x_{1} − x_{2}∥_{γ} = \sqrt{∫ |π^{1} − π^{2}|^{2}dγ} = \sqrt{∫ |x_{1} − x_{2}|^{2}dγ(x_{1}, x_{2})}.

The set Opt(μ_{0}, μ_{1}) is therefore defined as

(2.1) γ ∈ Opt(μ_{0}, μ_{1})  ⇔  γ ∈ Adm(μ_{1}, μ_{2}) and ∥x_{1} − x_{2}∥_{γ} = W (μ_{0}, μ_{1}).

In the following we will extensively use the simple and crucial inequality

(2.2)  W (T_{#}μ, S_{#}μ) ≤ ∥T − S∥_{μ},

which is a consequence of (T, S)_{#}μ ∈ Adm(T_{#}μ, S_{#}μ).

Theorem 2.2 (W is a distance). The map (μ, ν) → W (μ, ν) is a distance on 𝒫_{2}(ℝ^{d}).

12 2. THE DISTANCE W AND ITS GEOMETRY

Proof. It is clear that W is symmetric. Considering the plan (Id, Id)_{#}μ we get that W (μ, μ) = 0. Conversely if W (μ, ν) = 0, then there exists a plan γ ∈ Adm(μ, ν) such that ∫ |x_{1} −x_{2}|^{2}dγ = 0: this means that the maps π^{1}, π^{2} are equal in L_{γ}^{2}, and therefore μ = π_{#}^{1}γ = π_{#}^{2}γ = ν. To prove the triangular inequality we need Dudley’s lemma. Consider three measures μ_{1}, μ_{2}, μ_{3} and two optimal plans γ^{1,2} ∈ Opt(μ_{1}, μ_{2}), γ^{2,3} ∈ Opt(μ_{2}, μ_{3}): then we know that there exists a 3-plan γ such that

\begin{matrix}π_{#}^{1,2}γ = γ^{1,2}, \\ π_{#}^{2,3}γ = γ^{2,3}. \end{matrix}

Since π_{#}^{1,3}γ ∈ Adm(μ_{1}, μ_{3}), the conclusion follows by

\begin{matrix}W (μ_{1}, μ_{3}) ≤ & \sqrt{∫ |x_{1} − x_{3}|^{2}dπ_{#}^{1,3}γ} = \sqrt{∫ |x_{1} − x_{3}|^{2}dγ} \\ ≤ & \sqrt{∫ |x_{1} − x_{2}|^{2}dγ} + \sqrt{∫ |x_{2} − x_{3}|^{2}dγ} \\ = & \sqrt{∫ |x_{1} − x_{2}|^{2}dπ_{#}^{1,2}γ} + \sqrt{∫ |x_{2} − x_{3}|^{2}dπ_{#}^{2,3}γ} \\ = & W (μ_{1}, μ_{2}) + W (μ_{2}, μ_{3}). \end{matrix}

To get that W is real valued, i.e. it never attains the value +∞ use the triangular inequality and the remark below. □

Remark 2.3 (Boundedness). For any μ ∈ 𝒫_{2}(ℝ^{d}) the plan (Id, 0)_{#}μ is the unique element of Adm(μ, δ_{0}) and therefore it is optimal. In particular we get that

W^{2}(μ, δ_{0}) = ∥x∥_{μ}^{2},

and that a subset of 𝒫_{2}(ℝ^{d}) is bounded if and only if it is 2-bounded.

We collect in the next propositions the first properties about convergence and compactness.

Proposition 2.4. Let (μ_{n}^{1}), (μ_{n}^{2}) be two sequences in 𝒫_{2}(ℝ^{d}) weakly converging in duality with C_{b} to μ^{1}, μ^{2} respectively. Then it holds

W (μ^{1}, μ^{2}) ≤ \underline{lim}_{n→+∞} W (μ_{n}^{1}, μ_{n}^{2}).

Proof. Without loss of generality we can assume that the \underline{lim} is a limit. Choosing γ_{n} ∈ Opt(μ_{n}^{1}, μ_{n}^{2}), proposition 1.3 ensures that this sequence is tight and so from every subsequence we can extract a subsequence (γ_{n_{k}}) weakly converging to some γ. By remark 1.1 we have γ ∈ Adm(μ^{1}, μ^{2}); since |x_{1} − x_{2}|^{2} is continuous and bounded from below we get

W (μ^{1}, μ^{2}) ≤ ∥x_{1} − x_{2}∥_{γ} ≤ \underline{lim}_{k} ∥x_{1} − x_{2}∥_{γ_{n_{k}}} .

□

Proposition 2.5. The space (𝒫_{2}(ℝ^{d}), W) is a complete and separable metric space. A sequence (μ_{n}) W −converges to μ if and only if it is 2-uniformly integrable and weakly convergent to μ in duality with C_{b}. A subset of 𝒫_{2}(ℝ^{d}) is relatively compact if and only if it is 2-uniformly integrable.

1. THE DISTANCE W 13

Proof. Let us first check the completeness. Choose a Cauchy sequence (μ_{n}) and observe that such a sequence is bounded and therefore tight. Find a subsequence (μ_{n_{k}}) weakly converging to some μ and observe that

lim_{n} W (μ_{n}, μ) ≤ lim_{n} \underline{lim}_{k} W (μ_{n}, μ_{n_{k}}) = 0.

Now suppose that W (μ_{n}, μ) → 0, we want to prove that μ_{n} weakly converge to μ and that it is 2-uniformly integrable. Being the sequence bounded, we can find a subsequence μ_{n_{k}} weakly converging to some ν: the proof we just did on Cauchy sequences weakly converging gives ν = μ, and since this result is independent on the subsequence chosen we get the weak convergence of the whole sequence μ_{n}. To prove the 2-uniform integrability use proposition 1.4 and remark 2.3 to get

∥x∥_{μ} = W (μ, δ_{0}) = lim_{n} W (μ_{n}, δ_{0}) = lim_{n→∞} ∥x∥_{μ_{n}}.

To show that weak convergence and 2-uniform integrability imply W − convergence choose plans γ_{n} ∈ Opt(μ_{n}, μ) and observe that proposition 1.5 ensures the 2-uniform integrability of the sequence γ_{n}. Choose a subsequence γ_{n_{k}} weakly converging to some γ and note that proposition 1.11 ensures that γ ∈ Opt(μ, μ), and therefore γ = (Id, Id)_{#}μ, from which

lim_{k→∞} W (μ, μ_{n_{k}}) = lim_{k→∞} ∥x_{1} − x_{2}∥_{γ_{n_{k}}} = ∥x_{1} − x_{2}∥_{γ} = 0.

As usual, this is independent on the subsequence chosen and so we get W (μ, μ_{n}) → 0. The characterization of relatively compact sets is now straightforward. In order to get a countable dense subset consider the set of convex combinations of Dirac masses at rational points with rational coefficients. □

It is important to note that the space (𝒫_{2}(ℝ^{d}), W) is not locally compact: there exists no measure (not even one with compact support) with a compact neighborhood. This is essentially due to the fact that 2-boundedness and 2-uniform integrability are different concepts. In order to give an explicit example fix μ ∈ 𝒫_{2}(ℝ^{d}), ε > 0 and consider the measures

μ_{n} := (1 − \frac{1}{n^{2}} ) μ + \frac{1}{n^{2}} δ_{x_{n}},

where x_{n} satisfies |x_{n}| = nε − ∥x∥_{μ}. We claim that all of these measures belong to the ball with centre in μ and radius ε and that there is no subsequence convergent w.r.t. W. For the first part of the assertion observe that the plan

γ_{n} := (Id, Id)_{#} ((1 − \frac{1}{n^{2}} )μ) + \frac{1}{n^{2}} (Id, C_{x_{n}})_{#}μ,

belongs to Adm(μ, μ_{n}) (where C_{x} is the constant function on ℝ^{d} whose value is x) and calculate

W (μ, μ_{n}) ≤ ∥x_{1} − x_{2}∥_{γ_{n}} = \frac{1}{n}∥x − x_{n}∥_{μ} ≤ \frac{1}{n} (∥x∥_{μ} + |x_{n}|) = ε.

The second part follows from

\underline{lim}_{n→∞} ∫_{ℝ^{d}\B_{R}} |x|^{2}dμ_{n} ≥ \underline{lim}_{n→∞} \frac{|x_{n}|^{2}}{n^{2}} = ε^{2}.

2. Interpolation and geodesics

The interpolation between measures we define in this section is a fundamental tool for the next chapters: as we will see it is the natural interpolation that arises in the study of Wasserstein distance.

Definition 2.6. Let μ_{0}, μ_{1} ∈ 𝒫_{2}(ℝ^{d}) be two measures and let γ ∈ Adm(μ_{0}, μ_{1}) be a plan. The interpolation from μ_{0} to μ_{1} through γ is the curve [γ](t) : [0, 1] → 𝒫_{2}(ℝ^{d}) defined as

(2.3)  [γ](t) := ((1 − t)π^{1} + tπ^{2})_{#}γ.

Observe that [γ](0) = μ_{0} and [γ](1) = μ_{1}.

It may be noticed that the measures μ_{0}, μ_{1} are not needed in the definition, since the curve depends only on the plan γ, however we kept them because in the applications the extreme points of the curve are usually given and one has to look for a suitable admissible plan to get the right interpolation curve.

If γ = (Id, T)_{#}μ_{0} is induced by a map equation (2.3) reduces to

(2.4)  [γ](t) := ((1 − t)Id + tT )_{#}μ_{0}.

Proposition 2.7. For any μ_{0}, μ_{1} ∈ 𝒫_{2}(ℝ^{d}) and any γ ∈ Adm(μ_{0}, μ_{1}) the curve t → [γ](t) is Lipschitz with constant less or equal than the cost of γ.

Proof. It is sufficient to observe that for every 0 ≤ s < t ≤ 1 the plan

γ_{s,t} := ((1 − s)π^{1} + sπ^{2}, (1 − t)π^{1} + tπ^{2})_{#}γ

belongs to Adm([γ](s), [γ](t)) and that

(2.5)  W^{2}([γ](s), [γ](t)) ≤ ∥x_{1} − x_{2}∥_{γ_{s,t}}^{2} = (t − s)^{2}∥x_{1} − x_{2}∥_{γ}^{2}.
□

It is interesting to observe that the usual linear interpolation μ_{t} := (1 − t)μ_{0} + tμ_{1} still defines a continuous curve (see proposition 2.5), but in general this curve is not absolutely continuous. For istance choose μ_{0} := δ_{0} and μ_{1} := δ_{1}: it is not difficult to check that

W (μ_{t}, μ_{s}) = \sqrt{|t − s|}.

Observe that with the same choice of μ_{0}, μ_{1} the unique admissible plan is δ_{(0,1)} and it gives the curve t → δ_{t}, which is an “horizontal” interpolation between the two deltas which is radically different from the usual “vertical” one. Proposition 2.7 suggests that for γ ∈ Opt(μ_{0}, μ_{1}) the curve γ[t] is a geodesic from μ_{0} to μ_{1}. The next theorem shows that this is actually the case, and that every constant speed geodesic comes from this construction.

Theorem 2.8. Let μ, ν ∈ 𝒫_{2}(ℝ^{d}) be two measures. Then a curve μ_{t} : [0, 1] → 𝒫_{2}(ℝ^{d}) is a constant speed geodesic from μ to ν if and only if μ_{t} = [γ](t) for some optimal plan γ ∈ Opt(μ, ν).

Proof. Suppose that γ ∈ Opt(μ, ν) and note that equation (2.5) together with equality ∥x_{1} − x_{2}∥_{γ} = W (μ, ν) gives that [γ](t) is a constant speed geodesic (it cannot happen that the inequality in (2.5) is strict for some s < t, otherwise there would be a contradiction applying the triangular inequality to the measures μ = μ_{0}, μ_{s}, μ_{t}, μ_{1} = ν).

The converse implication is more difficult. Suppose μ_{0} = μ, μ_{1} = ν, fix 0 < t < 1 and choose plans γ_{0,t}, γ_{t,1} in Opt(μ_{0}, μ_{t}), and Opt(μ_{t}, μ_{1}) respectively. In order to let the proof be more clear we will use different names for the variables of different measures: x_{1}, x_{2}, x_{3} will be the variables of μ_{0}, μ_{t}, μ_{1} respectively.

Use Dudley’s lemma to find a 3-plan γ such that

π_{#}^{1,2}γ = γ_{0,t},
π_{#}^{2,3}γ = γ_{t,1},

and observe that it holds

(2.6) W (μ_{0}, μ_{1}) ≤ ∥x_{1} − x_{3}∥_{γ} ≤ ∥x_{1} − x_{2}∥_{γ} + ∥x_{2} − x_{3}∥_{γ} = W (μ_{0}, μ_{t}) + W (μ_{t}, μ_{1}) = W (μ_{0}, μ_{1}),

which implies in particular that π_{#}^{1,3}γ ∈ Opt(μ_{0}, μ_{1}). Our goal is to prove that μ_{t} = [π_{#}^{1,3}γ](t). Since all the inequalities in (2.6) are equalities and the L^{2} norm is strictly convex, we get that there exists some λ > 0 such that x_{1} − x_{2} = λ(x_{2} − x_{3}) for γ−a.e. (x_{1}, x_{2}, x_{3}). Moreover, since W (μ_{0}, μ_{t}) = \frac{t}{1−t}W (μ_{t}, μ_{1}) we have λ = \frac{t}{1−t} and so

(2.7)  x_{3} = x_{2} + \frac{1 − t}{t}(x_{2} − x_{1}),  γ − a.e. (x_{1}, x_{2}, x_{3}).

This equation implies that the 3-plan γ is actually a function of the 2-plan γ_{0,t}, more explicitly it holds

(2.8)  γ = (π^{1}, π^{2}, \frac{1}{t}π^{2} − \frac{1 − t}{t}π^{1})_{#} γ_{0,t}.

Since it holds π_{#}^{2,3}γ = γ_{t,1} we get that γ_{t,1} is a function of γ_{0,t}: since it was chosen independently we obtain that γ_{t,1} is unique. Arguing in similar way we get that γ_{0,t} is unique, too.

Up to now we know that [π_{#}^{1,3}γ](t) = μ_{t} (it follows from (2.8)) and we want to prove that the same result holds for t′ ≠ t. Suppose t′ < t (the other case is similar) and consider the curve s → μ_{st}: it is clearly a constant speed geodesic from μ_{0} to μ_{t}. Choosing s = t′/t, a plan γ_{0,t′} ∈ Opt(μ_{0}, μ_{t′}) (actually the plan) and arguing as before, the uniqueness of the optimal plan for (μ_{0}, μ_{t}) gives the formula

(2.9)  γ_{0,t} = (π^{1}, \frac{1}{s}π^{2} − \frac{1 − s}{s}π^{1})_{#} γ_{0,t′}.

We conclude with the following calculation

\begin{matrix}[γ](t′) = & [γ](st) = ((1 − st)π^{1} + stπ^{2})_{#}γ \\ = & ((1 − st)π^{1} + s(π^{2} − (1 − t)π^{1}))_{#}γ_{0,t} \\ = & ((1 − s)π^{1} + sπ^{2})_{#}γ_{0,t} = π_{#}^{2}γ_{0,t′} = μ_{t′}. \end{matrix}

□

The previous proposition characterizes geodesics parametrized in [0, 1], that is geodesics whose (constant) velocity is equal to the distance between the extreme measures. If we want to drop the restriction to [0, 1] it is sufficient to observe that μ_{t} : [0, \overline{t}] → 𝒫_{2}(ℝ^{d}) is a constant speed geodesic from μ to ν if and only if t → μ_{t\overline{t}} is a constant speed geodesic parametrized on [0, 1], that is, if and only if μ_{t\overline{t}} = [γ](t) for some optimal plan γ ∈ Opt(μ, ν), or, equivalently, iff

μ_{t} = [γ](t/\overline{t}) = ((1 − \frac{t}{\overline{t}})π^{1} + \frac{t}{\overline{t}}π^{2})_{#} γ.

Introducing the plan

(2.10)  γ\overline{t} := (π^{1}, π^{1} + \frac{π^{2} − π^{1}}{\overline{t}} )_{#} γ,

the previous equation may be written as

μ_{t} = [γ\overline{t}](t).

Since formula (2.10) is invertible, we get

γ = (π^{1}, π^{1} + \overline{t}(π^{2} − π^{1}))_{#} γ\overline{t},

and we obtain the following result:

Corollary 2.9. Let μ ∈ 𝒫_{2}(ℝ^{d}) be a measure. Then t → μ_{t} is a constant speed geodesic starting from μ parametrized in some right neighborhood of 0 if and only if μ_{t} = [γ](t) for some plan γ such that

\overline{γ} := (π^{1}, π^{1} + \overline{t}(π^{2} − π^{1}))_{#} γ is optimal for some \overline{t} > 0.

Moreover for t < \overline{t} the unique optimal plan from μ to μ_{t} is given by

(π^{1}, (1 − t)π^{1} + tπ^{2})_{#}γ = (π^{1}, π^{1} + \frac{t}{\overline{t}}(π^{2} − π^{1}))_{#} \overline{γ}.

Remark 2.10 (The regular case). Suppose that the measure μ is regular. Then theorems 2.8 and 1.15 ensure that for any ν ∈ 𝒫_{2}(ℝ^{d}) there exists a unique constant speed geodesic μ_{t} : [0, 1] → 𝒫_{2}(ℝ^{d}) from μ to ν, and this geodesic is given by (see also formula (2.4)):

μ_{t} = (Id + t(T_{μ}^{ν} − Id))_{#}μ.

Moreover the previous corollary gives that t → μ_{t} is a constant speed geodesic starting from μ parametrized in some right neighborhood of 0 if and only if

(2.11)  μ_{t} = (Id + \frac{t}{\overline{t}}(T − Id))_{#} μ,

3. THE CURVATURE OF (𝒫_{2}(ℝ^{d}), W) 17

for some optimal map T and some \overline{t} > 0. In this case it holds

(2.12)  T_{μ}^{μ_{t}} = Id + \frac{t}{\overline{t}}(T − Id), for t < \overline{t}.

In the next proposition we collect the first properties of geodesics.

Proposition 2.11. Let μ_{t} : [0, 1] → 𝒫_{2}(ℝ^{d}) be a geodesic.

i) For any t ∈ (0, 1) the sets Opt(μ_{0}, μ_{t}), Opt(μ_{t}, μ_{1}) contain only one element. Both of the optimal plan are induced by maps from μ_{t}. Those maps are Lipschitz with constant at most \frac{1}{1−t} and \frac{1}{t} respectively.

ii) The range of two different geodesics with the same extreme points are disjoint

(except for the extreme points themselves, of course). iii) If either μ_{0} or μ_{1} is regular then the geodesic t → μ_{t} is unique. Moreover in

this case μ_{t} is regular for every t ∈ (0, 1).

Proof. To prove i) recall that we already observed that for a given geodesic γ[t], 0 ≤ t ≤ 1 (where γ is an optimal plan) and for a fixed 0 < t < 1 there exists only one optimal plan between γ[0] and γ[t] given by γ_{0,t} := (π^{1}, (1 − t)π^{1} + tπ^{2})_{#}γ. To prove that γ_{0,t} is induced by a Lipschitz map from γ[t] observe that any point (z, w) in its support may be written as (x, (1 − t)x + ty) for some point (x, y) in the support of γ. Choose now any couple of points (z_{1}, w_{1}), (z_{2}, w_{2}) in the support of γ[t] and find points (x_{1}, y_{1}), (x_{2}, y_{2}) in the support of γ such that (z_{i}, w_{i}) = (x_{i}, (1 − t)x_{i} + ty_{i}), i = 1, 2. Use the monotonicity property of the support of γ to get

\begin{matrix}(1 − t)^{2}|z_{1} − z_{2}|^{2} ≤ & ⟨(1 − t)(x_{1} − x_{2}), (1 − t)(x_{1} − x_{2}) + t(y_{1} − y_{2})⟩ \\ ≤ & (1 − t)|x_{1} − x_{2}||(1 − t)(x_{1} − x_{2}) + t(y_{1} − y_{2})| \\ = & (1 − t)|z_{1} − z_{2}||w_{1} − w_{2}|. \end{matrix}

Statement ii) is another simple consequence of the proof of the previous proposition: indeed we showed that it is possible to recover an entire geodesic by knowing only one of its internal points. It is therefore impossible for two different geodesics to have a non empty intersection in their interior.

The first part of iii) was already noticed in remark 2.10. In order to prove the other assertion suppose that μ_{0} is regular, then we know that defining T_{t} := (1 − t)Id + tT_{μ_{0}}^{μ_{1}} the geodesic is given by μ_{t} = (T_{t})_{#}μ_{0}. From point (i) we know that T_{t} is invertible and that T_{t}^{−1} is Lipschitz (with constant less than (1 − t)^{−1}). Now let C be a H^{n−1} rectifiable set, since (T_{t})^{−1} is Lipschitz we know that (T_{t})^{−1}(C) is still a H^{n−1} rectifiable set, and so from μ_{t}(C) = μ_{0}((T_{t})^{−1}(C)) = 0 we conclude. □

3. The curvature of (𝒫_{2}(ℝ^{d}), W)

Here we begin a deeper analysis of the geometry of the space (𝒫_{2}(ℝ^{d}), W). The main result of this section is the semiconcavity inequality of W^{2} along geodesics (see inequality (2.16)).

In a Euclidean setting the unique geodesic from x_{0} to x_{1} is given by t → x_{t} := (1 − t)x_{0} + tx_{1}, a simple calculation shows that for any y it holds

(2.13)  |y − x_{t}|^{2} = (1 − t)|y − x_{0}|^{2} + t|y − x_{1}|^{2} − t(1 − t)|x_{0} − x_{1}|^{2}.

If we replace |x − y| with d(x, y) the objects involved make sense even in an arbitrary metric space with geodesics (in short a length space). Such a space is called positively curved (or P C space) in the sense of Aleksandrov if the inequality ≥ holds for any constant speed geodesic t → x_{t} and every y ∈ E, i.e.

(2.14)  d^{2}(y, x_{t}) ≥ (1 − t)d^{2}(y, x_{0}) + td^{2}(y, x_{1}) − t(1 − t)d^{2}(x_{0}, x_{1}).

and it is called non positively curved (or NP C space) if ≤ always holds.

The space (𝒫_{2}(ℝ^{d}), W) is a P C space, the proof is a simple consequence of the following variant of Dudley’s lemma.

Lemma 2.12. Let γ ∈ 𝒫_{2}(ℝ^{2d}) be a plan, t ∈ [0, 1] a fixed point, and γ_{t} ∈ 𝒫_{2}(ℝ^{2d}) such that π_{#}^{1}γ_{t} = [γ](t). Then there exists a 3-plan μ_{t} satisfying

(2.15a)  π_{#}^{1,2}μ_{t} = γ,
(2.15b)  ((1 − t)π^{1} + tπ^{2}, π^{3})_{#}μ_{t} = γ_{t}.

Proof. Use Dudley’s lemma to find a 3-plan \overline{μ} such that

\begin{aligned}π_{#}^{1,2}\overline{μ} & = (π^{1}, (1 − t)π^{1} + tπ^{2})_{#}γ, \\ π_{#}^{2,3}\overline{μ} & = γ_{t}, \end{aligned}

and define

μ_{t} := (π^{1}, π^{1} + \frac{π^{2} − π^{1}}{t} , π^{3})_{#}\overline{μ}.

□

Proposition 2.13 ((𝒫_{2}(ℝ^{d}), W) is positively curved). Let μ_{0}, μ_{1}, ν ∈ 𝒫_{2}(ℝ^{d}) be three measures and γ ∈ Opt(μ_{0}, μ_{1}) be an optimal plan. Then

(2.16) W^{2}(ν, [γ](t)) ≥ (1 − t)W^{2}(ν, μ_{0}) + tW^{2}(ν, μ_{1}) − t(1 − t)W^{2}(μ_{0}, μ_{1}), ∀t ∈ [0, 1].

Proof. Fix t ∈ [0, 1] and find, using lemma 2.12, a 3-plan μ satisfying

\begin{aligned}π_{#}^{1,2}μ_{t} & = γ, \\ ((1 − t)π^{1} + tπ^{2}, π^{3})_{#}μ_{t} & ∈ Opt([γ](t), ν). \end{aligned}

Then it holds

\begin{matrix}W^{2}(ν, [γ](t)) = & ∥(1 − t)x_{1} + tx_{2} − x_{3}∥_{μ_{t}}^{2} \\ = & (1 − t)∥x_{1} − x_{3}∥_{μ_{t}}^{2} + t∥x_{2} − x_{3}∥_{μ_{t}}^{2} − t(1 − t)∥x_{1} − x_{2}∥_{μ_{t}}^{2} \\ ≥ & (1 − t)W^{2}(ν, μ_{0}) + tW^{2}(ν, μ_{1}) − t(1 − t)W^{2}(μ_{0}, μ_{1}). \end{matrix} (2.17)

□

The proof of the previous proposition allows to understand the case of equality in (2.16). Indeed suppose that for some t_{0} ∈ (0, 1) it holds

W^{2}([γ](t_{0}), ν) = (1 − t_{0})W^{2}(ν, μ_{0}) + t_{0}W^{2}(ν, μ_{1}) − t_{0}(1 − t_{0})W^{2}(μ_{0}, μ_{1}),

then equation (2.17) shows that there exists a 3-plan μ such that

(2.18a)  π_{#}^{1,2}μ = γ,
(2.18b)  ((1 − t_{0})π^{1} + t_{0}π^{2}, π^{3})_{#}μ ∈ Opt([γ](t_{0}), ν),
(2.18c)  π_{#}^{1,3}μ ∈ Opt(μ_{0}, ν),
(2.18d)  π_{#}^{2,3}μ ∈ Opt(μ_{1}, ν).

Checking the cyclical monotonicity of the supports it is not difficult to verify that conditions (2.18c), (2.18d) imply that for every t ∈ [0, 1] the plan ((1−t)π^{1}+tπ^{2}, π^{3})_{#}μ has to be optimal (see also the proof of 4.25), moreover since ((1−t)π^{1}+tπ^{2})_{#}μ = [γ](t) we get ((1 − t)π^{1} + tπ^{2}, π^{3})_{#}μ ∈ Opt([γ](t), ν) for any t ∈ [0, 1]. In conclusion equality holds for some internal point if and only if it holds for any t ∈ [0, 1]. Thus, last condition is satisfied if and only if there exists a 3-plan μ such that

\begin{aligned}π_{#}^{1,2}μ & = γ, \\ π_{#}^{1,3}μ & ∈ Opt(μ_{0}, ν), \\ π_{#}^{2,3}μ & ∈ Opt(μ_{1}, ν). \end{aligned} ; (2.19)

For regular measures the equations above reduce to

(2.20)  T_{μ_{0}}^{μ_{1}} ∘ T_{ν}^{μ_{0}} = T_{ν}^{μ_{1}}.

It is for these reasons that at the end of the first chapter we interpreted the fact that in general T_{μ_{0}}^{μ_{1}} ∘ T_{ν}^{μ_{0}} ≠ T_{ν}^{μ_{1}} as a manifestation of the curvature of 𝒫_{2}(ℝ^{d}).

Observe that equation (2.16) is equivalent to say that the function W^{2}(·, σ) is -1– concave along geodesics. We can then ask whether the same function is λ–convex for some λ. The answer is no, as shown by the following example.

Example 2.14. Let ν := \frac{1}{2}(δ_{(0,0)} + δ_{(0,−1)}), μ_{0} := \frac{1}{2}(δ_{(−2,1)} + δ_{(0,0)}) and μ_{1} := \frac{1}{2}(δ_{(2,1)} + δ_{(0,0)}). Then it is easy to see that there exists only one geodesic μ_{t} connecting μ_{0} to μ_{1}, namely μ_{t} := \frac{1}{2}(δ_{(−2+2t,1−t)} + δ_{(2t,t)}). Moreover it can be checked that the infimum of the optimal transport problem from ν to μ_{t} is always attained by one of the two maps T_{t}^{1}, T_{t}^{2} defined as

\begin{aligned}T_{t}^{1}(0, 0) & = (−2 + 2t, 1 − t), & T_{t}^{1}(0, −1) & = (2t, t), \\ T_{t}^{1}(0, 0) & = (2t, t), & T_{t}^{1}(0, −1) & = (−2 + 2t, 1 − t). \end{aligned}

Therefore the function t → W^{2}(μ_{t}, ν), equal to

(2.21)  W^{2}(μ_{t}, ν) = \frac{1}{2} min \{10t^{2} − 8t + 6, 10t^{2} − 12t + 8\},

has a concave cusp at t = \frac{1}{2}.

4. The weak topology of 𝒫_{2}(X)

In this section we introduce a useful tool: the weak topology (𝒫_{2}(X), τ). In the following whenever we write 𝒫_{2}(X) without an explicit reference to a topology we will always assume we are working with the strong topology induced by W.

Let us start recalling the definition of inductive limit of topologies. Let (𝒳_{n}, τ_{n}), n ∈ ℕ, be a sequence of topological spaces such that 𝒳_{n} ⊂ 𝒳_{n+1} with continuous inclusion for every n. The inductive limit of the topologies τ_{n} is the strongest topology τ on 𝒳 := ∪_{n}𝒳_{n} which let the inclusions ι_{n} : 𝒳_{n} ↪ 𝒳 be continuous for every n ∈ ℕ.

Proposition 2.15. It holds:

i) A subset A ⊂ 𝒳 belongs to τ if and only if

(2.22)  A ∩ 𝒳_{n} ∈ τ_{n}  ∀n ∈ ℕ.

The analogous property holds for closed sets, too. ii) Let x ∈ 𝒳 be a point and N be the lowest integer such that x ∈ 𝒳_{N}. A subset U ⊂ 𝒳 is a neighborhood of x if and only if U ∩ 𝒳_{n} is a neighborhood of x in (𝒳_{n}, τ_{n}) for every n ≥ N.

iii) τ is Hausdorff if and only if each τ_{n} is Hausdorff.

iv) Suppose that τ is Hausdorff. A sequence (x_{n}) ⊂ 𝒳 converges to some x ∈ 𝒳 if and only if there exists N such that x ∈ 𝒳_{N}, x_{n} ∈ 𝒳_{N} ∀n, and x_{n} → x in (𝒳_{N}, τ_{N}).

Proof. Let A ∈ τ be an open set. Since ι_{n}^{−1}(A) = A ∩ 𝒳_{n} and ι_{n} is continuous we have that A ∩ 𝒳_{n} is open for every n. Conversely the set identities

(⋃_{i∈I} A_{i}) ∩ 𝒳_{n} =  ⋃_{i∈I} (A_{i} ∩ 𝒳_{n})  ∀n ∈ ℕ,
(A_{1} ∩ A_{2}) ∩ 𝒳_{n} = (A_{1} ∩ 𝒳_{n}) ∩ (A_{2} ∩ 𝒳_{n})  ∀n ∈ ℕ,

imply that the set of A’s which satisfy (2.22) is a topology. The statement on closed sets follows from

(𝒳 \ A) ∩ 𝒳_{n} = 𝒳_{n} \ A, ∀n ∈ ℕ.

ii) and iii) are straightforward consequences of i). The if implication of iv) is obvious, so we turn to the only if. Let x_{n} → x be a converging sequence in 𝒳 , clearly it is enough to show that there exists M such that x_{n} ∈ 𝒳_{M} for every n. Suppose on the contrary that this is not the case, i.e. suppose that for every k there exists x_{n_{k}} ∉ 𝒳_{k}, without loss of generality we can assume that k → n_{k} is an increasing sequence and x_{n_{k}} ≠ x ∀k. Now simply observe that the complement of {x_{n_{k}}}_{k∈ℕ} is an open set (since its intersection with every 𝒳_{n} is the complement of a finite set) which contains x but contains none of the x_{n_{k}}. □

Observe that if one takes a sequence of sets A_{n} ⊂ 𝒳_{n} such that A_{n} ∈ τ_{n} for every n ∈ ℕ, it may happen that A := ∪_{n}A_{n} is not an open set in 𝒳 : indeed since A ∩ 𝒳_{n} = ∪_{1}^{n}A_{i} ∪ ((∪_{n+1}^{∞}A_{i}) ∩ x_{n}), and the first term may not belong to τ_{n}, this set may fail to be open.

4. THE WEAK TOPOLOGY OF 𝒫_{2}(X) 21

Let us now come back to our case. Define

𝒳_{n} := {μ ∈ 𝒫(X) : ∫ |x|^{2}dμ ≤ n},

and let τ_{n} be the usual weak topology on 𝒳_{n}. Clearly it holds 𝒫_{2}(X) = ∪𝒳_{n}.

Definition 2.16. The weak topology τ of 𝒫_{2}(X) is the inductive limit of (𝒳_{n}, τ_{n}).

Note that the definition makes sense because 𝒳_{n} ⊂ 𝒳_{n+1} and the inclusion is continuous.

The first properties of (𝒫_{2}(X), τ) are collected below.

• The topology τ is not the weak topology on 𝒫_{2}(X) induced by the duality with continuous and bounded functions. Indeed by proposition 2.15 we have that μ_{n} →^{τ} μ if and only if μ_{n} ⇀ μ and μ_{n} is 2-bounded.

• Closed balls in (𝒫_{2}(X), W) are compact and sequentially compact in (𝒫_{2}(X), τ). Indeed the previous statement implies that balls are sequentially compact, and to get compactness observe that for any closed ball B there exists N such that B ⊂ 𝒳_{N}: since 𝒳_{N} is compact the conclusion follows.

• (𝒫_{2}(X), τ) is not induced by a distance. Indeed ii) of proposition 2.15 suggests that (𝒫_{2}(X), τ) does not satisfy the first axiom of numerability. The following is an explicit counterexample.

Example 2.17. Let X := ℝ, μ := δ_{0}, μ_{n} := (1 − \frac{1}{n^{2}} )δ_{0} + \frac{1}{n^{2}} δ_{n} and μ_{n,m} := (1 − \frac{1}{n^{2}} − \frac{1}{m^{2}} )δ_{0} + \frac{1}{n^{2}} δ_{n} + \frac{1}{m^{2}} δ_{nm}. Then μ_{n} →^{τ} μ and for a fixed n it holds μ_{n,m} →^{τ} μ_{n} (since ∫ |x|^{2}dμ_{n,m} = 1 + n^{2} for every m), but we cannot find a sequence of elements of the form μ_{n,m} which τ−converge to μ.

• (𝒫_{2}(X), τ) is not locally sequentially compact. Indeed choose an open neighborhood U of δ_{0} and observe that each sequence (μ_{n}) ⊂ (U ∩ 𝒳_{n}) \ 𝒳_{n−1} does not have any converging subsequence (note that it eventually holds U ∩ 𝒳_{n−1} ⊊ U ∩ 𝒳_{n}).

• W is sequentially lower semicontinuous w.r.t. τ. Indeed for any couple of sequences μ_{n}, ν_{n} τ−converging to μ, ν respectively and any choice of optimal plans γ_{n} ∈ Opt(μ_{n}, ν_{n}) we have by proposition 1.5 that (γ_{n}) is bounded in (𝒫_{2}(ℝ^{2d}), W). Therefore it is sequentially compact w.r.t. τ and for every subsequence (γ_{n_{k}}) converging to a certain γ we have γ ∈ Adm(μ, ν) and therefore

(2.23)  W (μ, ν) ≤ ∥x_{1} − x_{2}∥_{γ} ≤ \underline{lim}_{k→∞} ∥x_{1} − x_{2}∥_{γ_{n_{k}}} = \underline{lim}_{k→∞} W (μ_{n_{k}}, ν_{n_{k}}).

Being the result independent on the subsequence chosen we get the claim. In the next propositions we analyze the relationship between convergence of mea-

sures, weak or strong, and convergence of optimal plans, in the same spirit of proposition 1.11. The first proposition below shows that the plan γ of equation (2.23) is actually optimal for the couple μ, ν.

Proposition 2.18. Let μ_{n}, ν_{n}, μ, ν ∈ 𝒫_{2}(ℝ^{d}) be given measures such that μ_{n} →^{τ} μ, ν_{n} →^{τ} ν, and let γ_{n} ∈ Opt(μ_{n}, ν_{n}) be any choice of optimal plans. Then the sequence (γ_{n}) is relatively compact in (𝒫_{2}(ℝ_{1}^{d} × ℝ_{2}^{d}), τ) and every τ accumulation point belongs to Opt(μ, ν).

Proof. The relative compactness follows from proposition 1.5 and remark 2.3, while the optimality of the limit plan is a consequence of proposition 1.11. □

The analogous statement for strong convergence holds with a weaker assumption on the plans γ_{n}:

Proposition 2.19. Let μ_{n}, ν_{n}, μ, ν ∈ 𝒫_{2}(ℝ^{d}) be given measures such that μ_{n} →^{W} μ, ν_{n} →^{W} ν, and let γ_{n} ∈ Adm(μ, ν) be any choice of admissible plans such that

\overline{lim}_{n→∞} ∥x_{1} − x_{2}∥_{γ_{n}} ≤ W (μ, ν).

Then the sequence γ_{n} is relatively compact in (𝒫_{2}(ℝ_{1}^{d} × ℝ_{2}^{d}), W) and any W accumulation point belongs to Opt(μ, ν).

Proof. The relative compactness is a consequence of propositions 1.5, 2.5, while the optimality of the limit plan follows by

W (μ, ν) ≤ ∥x_{1} − x_{2}∥_{γ} ≤ \underline{lim}_{k→∞} ∥x_{1} − x_{2}∥_{γ_{n_{k}}} ≤ W (μ, ν).

□

5. Convergence of maps

In the following we will usually need to find the limit map, in a certain sense, of a sequence of maps T_{n} ∈ L^{2}(ℝ^{d}, ℝ^{d′}; μ_{n}) when the measures μ_{n} are different. The result will be a couple (T, μ) with T ∈ L^{2}(ℝ^{d}, ℝ^{d′}; μ). We are going to describe two type of convergence which are in some sense similar to the strong and to the weak convergence in L^{2}.

Definition 2.20 (Weak and strong convergence of maps). Let (T, μ), (T_{n}, μ_{n}), n ∈ ℕ, be given couples such that T_{n} ∈ L^{2}(ℝ^{d}, ℝ^{d′}; μ_{n}) and T ∈ L^{2}(ℝ^{d}, ℝ^{d′}; μ). We say that (T_{n}, μ_{n}) weakly converge to (T, μ), and write (T_{n}, μ_{n}) →^{τ} (T, μ), if

μ_{n} →^{τ} μ, (2.24)

T_{n}μ_{n} ⇀ T μ, (2.25)

sup_{n∈ℕ}∥T_{n}∥_{μ_{n}} < +∞, (2.26)

where the convergence in (2.25) is in duality with C_{c}, and, in (2.26), ∥T_{n}∥_{μ_{n}} stands for the L^{2} norm of T_{n} as element of L^{2}(ℝ^{d}, ℝ^{d′}, μ_{n}). We say that (T_{n}, μ_{n}) strongly converge (or simply converge) to (T, μ), and write (T_{n}, μ_{n}) ↠ (T, μ), if

μ_{n} →^{τ} μ, (2.27)

T_{n}μ_{n} ⇀ T μ, (2.28) ∥T_{n}∥_{μ_{n}} → ∥T ∥_{μ}. (2.29)

Observe that if μ_{n} = μ for every n these definitions reduce to the usual weak and strong convergence in L^{2}(ℝ^{d}, ℝ^{d′}, μ). It is possible to define appropriate topologies which induce these distances with the same ideas used to construct τ, but we do not investigate them.

In the following when we write a couple (T, μ) it will be always intended that T is an L^{2} map w.r.t. μ with values in some Euclidean space ℝ^{d′}, moreover we will sometimes write T_{n} ↠ T (or T_{n} →^{τ} T) omitting the explicit reference to the measures μ_{n}, μ if there is no risk of confusion. Before studying the first properties of weak and strong convergence of maps we need the following definition.

Definition 2.21 (Barycentric projection). Let γ ∈ 𝒫_{2}(ℝ_{1}^{d} × ℝ_{2}^{d′}) be a plan and let dγ = dπ_{#}^{1}γ ⊗ dγ_{x_{1}} be its disintegration w.r.t. its first marginal. The barycentric projection of γ w.r.t. π_{#}^{1}γ is the function ℬ(γ)(x_{1}) : ℝ^{d} → ℝ^{d′} defined as

ℬ(γ)(x_{1}) := ∫ x_{2}dγ_{x_{1}}.

Observe that since

(2.30) ∥ℬ(γ)∥_{π_{#}^{1}γ}^{2} = ∫ |ℬ(γ)(x_{1})|^{2}dπ_{#}^{1}γ = ∫ |∫ x_{2}dγ_{x_{1}} |^{2} dπ_{#}^{1}γ

≤ ∫ |x_{2}|^{2}dγ_{x_{1}}dπ_{#}^{1}γ = ∫ |x_{2}|^{2}dγ < +∞,

ℬ(γ) is a well defined function belonging to L^{2}(ℝ^{d}, ℝ^{d′}; π_{#}^{1}γ). The same equation implies that equality holds if and only if γ_{x_{1}} = δ_{ℬ(γ)(x_{1})} for π_{#}^{1}γ-a.e. x_{1}, i.e.

(2.31)  ∥ℬ(γ)∥_{π_{#}^{1}γ} = ∥x_{2}∥_{γ} ⇔ γ = (Id, ℬ(γ))_{#}(π_{#}^{1}γ).

The barycentric projection is also characterized by the following formula

(2.32)  ∫ ⟨φ(x_{1}), x_{2}⟩dγ = ∫ ⟨φ(x_{1}), ℬ(γ)(x_{1})⟩dπ_{#}^{1}γ, ∀φ ∈ C_{c}^{∞}(ℝ^{d}, ℝ^{d′}).

Proposition 2.22. Let (T_{n}, μ_{n}), (T, μ), n ∈ ℕ, be given couples. Then

(2.33)  (Id, T_{n})_{#}μ_{n} →^{τ} (Id, T)_{#}μ ⇒ (T_{n}, μ_{n}) →^{τ} (T, μ).

Conversely if (T_{n}, μ_{n}) →^{τ} (T, μ) then the family of plans (Id, T_{n})_{#}μ_{n} is 2-bounded and for any weak accumulation point γ it holds

(2.34)  ℬ(γ) = T.

Proof. Let us start with (2.33). Since π_{#}^{1}(Id, T_{n})_{#}μ_{n} = μ_{n} and ∥T_{n}∥_{μ_{n}}^{2} = ∫ |x_{2}|^{2}d(Id, T_{n})_{#}μ_{n} conditions (2.24) and (2.26) are fulfilled. Moreover choosing test functions of the form ψ(x_{1}, x_{2}) = ⟨φ(x_{1}), x_{2}⟩ we easily obtain (2.25). On the other hand the same choice of ψ and equation (2.32) yield to (2.34). □

Observe that this proposition yields the inequality

(2.35)  (T_{n}, μ_{n}) →^{τ} (T, μ) ⇒ ∥T ∥_{μ} ≤ \underline{lim}_{n→∞} ∥T_{n}∥_{μ_{n}}.

Indeed, from equation (2.30) we get

∥T ∥_{μ} = ∥ℬ(γ)∥_{μ} ≤ ∥x_{2}∥_{γ} ≤ \underline{lim}_{n→∞} ∥x_{2}∥_{γ_{n}} = \underline{lim}_{n→∞} ∥T_{n}∥_{μ_{n}}.

Therefore as in the usual case of Hilbert spaces the difference between strong and weak convergence is that in the first case there is no loss of norm.

Proposition 2.23. The following statements are equivalent. i)

(2.36)  (T_{n}, μ_{n}) ↠ (T, μ).

ii) μ_{n} →^{τ} μ and for any choice of γ_{n} ∈ Opt(μ_{n}, μ) it holds

(2.37)  ∥T_{n}(x_{1}) − T (x_{2})∥_{γ_{n}} → 0.

iii) For any continuous function φ with linear growth and values in some Euclidean space, it holds

(2.38)  (φ ∘ T_{n}, μ_{n}) ↠ (φ ∘ T, μ).
iv) ∥T_{n}∥_{μ_{n}} → ∥T ∥_{μ} and
(2.39)  (Id, T_{n})_{#}μ_{n} →^{τ} (Id, T)_{#}μ.

Proof. iv) ⇒ i) follows easily from (2.33). Conversely suppose that (Id, T_{n_{k}})_{#}μ_{n_{k}} →^{τ} γ and use the other implication of proposition 2.22 to get

∥T ∥_{μ} = ∥ℬ(γ)∥_{μ} ≤ ∥x_{2}∥_{γ} ≤ \underline{lim}_{k→∞} ∥x_{2}∥_{γ_{n_{k}}} = \underline{lim}_{k→∞} ∥T_{n_{k}}∥_{μ_{n_{k}}} = ∥T ∥_{μ}.

Therefore equation (2.31) gives

γ = (Id, ℬ(γ))_{#}μ = (Id, T)_{#}μ.

Now we turn to i) ⇒ ii): note that proposition 1.11 implies γ_{n} →^{τ} (Id, Id)_{#}μ. Now fix ψ ∈ C_{b}(ℝ^{d}, ℝ^{d′}) and observe that since ψ(x_{1}) − ψ(x_{2}) ∈ C_{b}(ℝ_{1}^{d} × ℝ_{2}^{d}, ℝ^{d′}) it holds ∥ψ(x_{1}) − ψ(x_{2})∥_{γ_{n}} → 0, moreover

\overline{lim}_{n→∞} ∥T_{n} − ψ∥_{μ_{n}}^{2} = ∥T ∥_{μ}^{2} + ∥ψ∥_{μ}^{2} − 2 \overline{lim}_{n→∞} ∫ ⟨T_{n}, ψ⟩dμ_{n} = ∥T − ψ∥_{μ}^{2}.

Hence, we can conclude with

\overline{lim}_{n→∞} ∥T_{n}(x_{1}) − T (x_{2})∥_{γ_{n}}

≤ \overline{lim}_{n→∞} ∥T_{n} − ψ∥_{μ_{n}} + \overline{lim}_{n→∞} ∥ψ(x_{1}) − ψ(x_{2})∥_{γ_{n}} + ∥ψ − T ∥_{μ} = 2∥ψ − T ∥_{μ},

by letting ψ tend to T in L^{2}(ℝ^{d}, ℝ^{d′}, μ).

To show that ii) ⇒ i) begin observing that

∥|T_{n}∥_{μ_{n}} − ∥T ∥_{μ}| ≤ ∥T_{n}(x_{1}) − T (x_{2})∥_{γ_{n}} → 0,

to get (2.29). Then choose ψ ∈ C_{c}(ℝ^{d}, ℝ^{d′}) and note that since ∥ψ(x_{1}) − ψ(x_{2})∥_{γ_{n}} → 0 and ∥T_{n}(x_{1}) − T (x_{2})∥_{γ_{n}} → 0 equation (2.28) follows from

∫ ⟨ψ, T_{n}⟩dμ_{n} − ∫ ⟨ψ, T ⟩dμ = ∫ ⟨ψ(x_{1}), T_{n}(x_{1})⟩ − ⟨ψ(x_{2}), T (x_{2})⟩dγ_{n}(x_{1}, x_{2})

=∫ ⟨ψ(x_{1}), T_{n}(x_{1}) − T (x_{2})⟩dγ_{n}(x_{1}, x_{2}) +∫ ⟨ψ(x_{1}) − ψ(x_{2}), T (x_{2})⟩dγ_{n}(x_{1}, x_{2})→0.

Since it is clear that iii) ⇒ i) it remains to prove that i), ii), iv) ⇒ iii). We will show that if (T_{n}, μ_{n}), (T, μ) satisfy (2.37) then (φ ∘ T_{n}, μ), (φ ∘ T, μ) still satisfy (2.37). Since

|∫ φ_{1} ∘ T − φ_{2} ∘ T dμ| ≤ sup_{x} |φ_{1}(x) − φ_{2}(x)|,

it suffices to check (2.38) for Lipschitz functions φ. Under this hypothesis we have

∥φ(T_{n}(x_{1})) − φ(T (x_{2}))∥_{γ_{n}} ≤ Lip_{φ} ∥T_{n}(x_{1}) − T (x_{2})∥_{γ_{n}} → 0.

□

Remark 2.24. It is not hard to see that, with the same notation of the above proposition, if we assume W (μ_{n}, μ) → 0 as n → ∞, then the strong convergence of maps is equivalent to ∥T (x) − T_{n}(y)∥_{γ_{n}} → 0 as n → ∞, where γ_{n} ∈ Adm(μ, μ_{n}) is any choice of plans satisfying ∥x − y∥_{γ_{n}} → 0 as n → ∞.

Observe that if μ is regular equation (2.37) reads as follows

T_{n} ∘ T_{μ}^{μ_{n}} → T in L^{2}(ℝ^{d}, ℝ^{d′}, μ).

Corollary 2.25. Let (T_{n}, μ_{n}) be couples strongly converging to (T, μ). Then

(T_{n})_{#}μ_{n} → T_{#}μ, in 𝒫_{2}(ℝ^{d′}, W).

Moreover, if in addition μ_{n} → μ then

(Id, T_{n})_{#}μ_{n} → (Id, T)_{#}μ, in 𝒫_{2}(ℝ^{d+d′}, W).

Proof. It is a straightforward consequence of the previous proposition. □

A consequence of propositions 2.18, 2.19 is the close link between convergence of measures and convergence of optimal maps given below.

Proposition 2.26. Let μ_{n}, ν_{n}, μ, ν ∈ 𝒫_{2}(ℝ^{d}) be given measures such that there exists optimal maps T_{n} from μ_{n} to ν_{n} and T from μ to ν. Suppose moreover that μ_{n} →^{τ} μ,ν_{n} →^{τ} ν and that (Id, T)_{#}μ is the unique element of Opt(μ, ν). Then

(T_{n}, μ_{n}) →^{τ} (T, μ).

Proposition 2.27. Let μ_{n}, ν_{n}, μ, ν ∈ 𝒫_{2}(ℝ^{d}) be given measures such that μ_{n} →^{W} μ, ν_{n} →^{W} ν and (Id, T)_{#}μ is the unique element of Opt(μ, ν) for some map T. Then if T_{n} ∈ L^{2}(μ_{n}, ℝ^{d}) is any sequence satisfying (T_{n})_{#}μ_{n} = ν_{n} and

(2.40)  \overline{lim}_{n→∞} ∥T_{n} − Id∥_{μ_{n}} ≤ W (μ, ν),

it holds

(T_{n}, μ_{n}) ↠ (T, μ).

Proof. Proposition 2.26 is a simple consequence of proposition 2.18, 2.22, while proposition 2.27 follows from 2.19 and 2.23. □

Observe that the conditions on the given measures are fulfilled if they are regular; we will mostly use the previous propositions under this assumption.

The following lemma is a variant of the classical result in Hilbert spaces:

v_{n} → v, w_{n} ⇀ w ⇒ ⟨v_{n}, w_{n}⟩ → ⟨v, w⟩.

Lemma 2.28. Let (T_{n}, μ_{n}), (S_{n}, μ_{n}), n ∈ ℕ, be given couples such that all the maps have values in the same ℝ^{d′} and suppose that

(T_{n}, μ_{n}) ↠ (T, μ), (2.41)

(S_{n}, μ_{n}) →^{τ} (S, μ), (2.42)

for some measure μ and maps T, S. Then

(2.43)  lim_{n→∞} ∫ ⟨T_{n}, S_{n}⟩dμ_{n} = ∫ ⟨T, S⟩dμ.

Proof. Let γ_{n} ∈ Opt(μ_{n}, μ), n ∈ ℕ, be any choice of optimal plans and ψ ∈ C_{c}(ℝ^{d}, ℝ^{d′}) be a fixed function. Reasoning as in the proof of proposition 2.23, from (2.41) we know that

(2.44)  lim_{n→∞} ∥T_{n} − ψ∥_{μ_{n}} = ∥T − ψ∥_{μ},

which yield to

\begin{matrix}|∫ ⟨T_{n}, S_{n}⟩dμ_{n} − ∫ ⟨T, S⟩dμ| ≤ & |∫ ⟨T_{n}(x_{1}) − ψ(x_{1}), S_{n}(x_{1})⟩dγ_{n}| \\ + & |∫ ⟨ψ(x_{1}), S_{n}(x_{1})⟩ − ⟨ψ(x_{2}), S(x_{2})⟩dγ_{n}| \\ + & |∫ ⟨ψ(x_{2}) − T (x_{2}), S(x_{2})⟩dγ_{n}| . \end{matrix}

Taking the limit, using (2.44), (2.42) and letting C := sup_{n} ∥S_{n}∥_{μ_{n}} we get

\overline{lim}_{n→∞} |∫ ⟨T_{n}, S_{n}⟩dμ_{n} − ∫ ⟨T, S⟩dμ| ≤ 2C∥T − ψ∥_{μ},

and by the arbitrarieness of ψ the thesis follows. □

We will mostly apply this lemma in the following form:

Corollary 2.29. Let (γ_{n}) be a sequence of plans τ−converging to some γ such that

π_{#}^{1}γ_{n} →^{W} π_{#}^{1}γ.

Then

∫ ⟨x_{1}, x_{2}⟩dγ_{n} → ∫ ⟨x_{1}, x_{2}⟩dγ.

Proof. It is sufficient to apply the previous lemma with μ_{n} := γ_{n}, T_{n} := π^{1}, S_{n} := π^{2}. □

We conclude with the following proposition, which shows that bounded sequences of maps are weakly compact.

Proposition 2.30 (Weak compactness of maps). Let (μ_{n}) be a 2-bounded sequence of measures and let T_{n} ∈ L^{2}(ℝ^{d}, ℝ^{d′}, μ_{n}) be a sequence of maps such that ∥T_{n}∥_{μ_{n}} ≤ C < +∞. Then there exists a subsequence (T_{n_{k}}, μ_{n_{k}}) weakly converging to some couple (T, μ).

Proof. Since the sequence of measures is 2-bounded, we can extract a subsequence, not relabeled, which weakly converges to a measure μ. Now consider a countable set D ⊂ C_{c}^{∞}(ℝ^{d}, ℝ^{d′}) which is dense in L_{μ}^{2} and observe that it holds

\overline{lim}_{n→∞} |⟨T_{n}, φ⟩_{μ_{n}} | ≤ \overline{lim}_{n→∞} ∥T_{n}∥_{μ_{n}}∥φ∥_{μ_{n}} ≤ C∥φ∥_{μ},

for any φ ∈ D. Therefore we can extract a subsequence (T_{n_{k}}) such that there exists the limit L(φ) of k → ⟨T_{n_{k}}, φ⟩_{μ_{n_{k}}} for any φ ∈ D. The functional L can be clearly extended to a linear functional on the whole L_{μ}^{2} whose norm is controlled by C (by the density of D), therefore from the Rietz representation theorem we get the existence of a map T ∈ L_{μ}^{2} such that L(S) = ⟨T, S⟩_{μ} for any S ∈ L_{μ}^{2}. From the density of D in C_{c} we get that

lim_{n→∞} ⟨φ, T_{n}⟩_{μ_{n}} = ⟨φ, T ⟩_{μ}, ∀φ ∈ C_{c}(ℝ^{d}, ℝ^{d′}),

therefore the proof is complete. □

Remark 2.31. In the following we will often use, without explicit mention, the fact that for a τ−converging sequence of measures (μ_{n}) and a sequence of maps f_{n} ∈ L_{μ_{n}}^{2} such that sup_{n} ∥f_{n}∥_{μ_{n}} < ∞, in order to check the weak convergence of f_{n} it is sufficient to check the convergence in duality with functions in C_{c}^{∞}(ℝ^{d}). This is an easy consequence of the tightness of (μ_{n}) and the uniform bound of the norms of f_{n}.

CHAPTER 3 The regular tangent space

In this chapter we investigate the so-called Riemannian structure of (𝒫_{2}(ℝ^{d}), W). The ideas presented here were first studied by Otto in [46], and subsequentely made rigorous in [11]: the approach we present follows the one of the latter work. The first step will be the study of the continuity equation

(3.1)  \frac{d}{dt}μ_{t} + ∇ · (v_{t}μ_{t}) = 0,

which enables us to characterize absolutely continuous curves. As we will see, this will naturally lead to the introduction of the regular tangent space as

Tan_{μ}(𝒫_{2}(ℝ^{d})) := \overline{\{∇φ : φ ∈ C_{c}^{∞}(ℝ^{d})\}}^{L^{2}(μ,ℝ^{d})}.

This space is the object of analysis of the second section of the chapter.

1. Continuity equations and AC curves

Recall that a curve t → u_{t} on an interval I, with values in a metric space (E, d), is said to be absolutely continuous (AC) if there exists a function g ∈ L^{1}(I) such that

(3.2)  d(u_{t}, u_{s}) ≤ ∫_{t}^{s} g(r)dr, ∀t ≤ s, t, s ∈ I.

It is a well known fact (see for instance [13]) that for a given AC curve u(t) the limit

lim_{h→0} \frac{d(u_{t+h}, u_{t})}{h}

exists for a.e. t ∈ I and defines a fuction which belongs to L^{1}(I) and is minimal among the functions g satisfying (3.2). This function is usually denoted by t → |u̇_{t}| and called metric derivative, or metric speed.

The length L(u_{t}) of a curve t → u_{t} is defined as

L(u_{t}) := sup ∑_{i=0}^{n−i} d(u_{t_{i}}, u_{t_{i+1}}),

where the sup is taken among all n ∈ ℕ and partitions {t_{0} ≤ ⋯ ≤ t_{n}} of the inteval I. For an AC curve it is possible to show that

L(u_{t}) = ∫_{I} |u̇_{t}|dt.

In this section we will show that absolutely continuous curves in 𝒫_{2}(ℝ^{d}) are completely characterized by equation (3.1), meaning that:

• for every absolutely continuous curve [0, 1] ∋ t → μ_{t} ∈ 𝒫_{2}(ℝ^{d}) it is possible to find “velocity vectors” v_{t} such that ∥v_{t}∥_{μ_{t}} ≤ |μ̇_{t}| for a.e. t and (3.1) holds,

• conversely, every distributional solution of (3.1) with ∥v_{t}∥_{μ_{t}} ∈ L^{1}(0, 1) gives a curve μ_{t} which is a.e. equal to an absolutely continuous curve whose metric speed is bounded by ∥v_{t}∥_{μ_{t}}.

We will skip some technical details of the proofs, and refer to [11] for a more detailed analysis.

Before proving our main result we need a couple of lemmata.

Lemma 3.1 (Time rescaling). Let (μ_{t}, v_{t}) be a solution of (3.1), t → γ(t) a strictly increasing and absolutely continuous map and define \overline{μ}_{t} := μ_{γ(t)}, \overline{v}_{t} := γ′(t)v_{t}. Then (\overline{μ}_{t}, \overline{v}_{t}) is a solution of (3.1), too.

Proof. Straightforward. □

Proposition 3.2 (Uniqueness of solutions of continuity equation). Let σ_{t}, 0 ≤ t ≤ 1, be a family of signed measures solving

\frac{d}{dt}σ_{t} + ∇ · (v_{t}σ_{t}) = 0,

with σ_{0} ≤ 0,

∫_{0}^{1} ∫ |v_{t}|d|σ_{t}| < +∞

and

∫_{0}^{1} (|σ_{t}|(B) + sup_{B} |v_{t}| + Lip(v_{t}, B)) dt < +∞

for each compact set B ⊂ ℝ^{d}. Then σ_{t} ≤ 0 for any t ∈ [0, 1].

Proof. Fix ψ ∈ C_{c}^{∞}(ℝ^{d} × (0, 1)) with 0 ≤ ψ ≤ 1, R > 0, and a smooth cut-off function χ_{R}(·) = χ(·/R) ∈ C_{c}^{∞}(ℝ^{d}) such that 0 ≤ χ_{R} ≤ 1, |∇χ_{R}| ≤ 2/R, χ_{R} ≡ 1 on B_{R}(0) and χ_{R} ≡ 0 outside B_{2R}(0). We define w_{t} so that w_{t} = v_{t} on B_{2R}(0) × (0, 1), w_{t} = 0 if t ≠ [0, 1] and

sup_{ℝ^{d}}|w_{t}| + Lip(w_{t}, ℝ^{d}) ≤ sup_{B_{2R}(0)} |v_{t}| + Lip(v_{t}, B_{2R}(0)) ∀t ∈ [0, 1].

Let w_{t}^{ε} be obtained from w_{t} by a double mollification with respect to the space and time variables: notice that w_{t}^{ε} satisfy

sup_{ε∈(0,1)} ∫_{0}^{1} (sup_{ℝ^{d}}|w_{t}^{ε}| + Lip(w_{t}^{ε}, ℝ^{d})) dt < +∞.

We now build with the method of characteristics a smooth solution φ^{ε} : ℝ^{d} ×[0, 1] → ℝ of the PDE

\frac{d}{dt}φ^{ε} + ⟨w_{t}^{ε}, ∇φ^{ε}⟩ = ψ in ℝ^{d} × (0, 1), φ^{ε}(x, 1) = 0 ∀x ∈ ℝ^{d}.

1. CONTINUITY EQUATIONS AND AC CURVES 31

It is not difficult to check (we skip the details) that −1 ≤ φ^{ε} ≤ 0 and that |∇φ^{ε}| is uniformly bounded w.r.t. ε, t and x.

We insert now the test function φ^{ε}χ_{R} in the continuity equation and take into account that σ_{0} ≤ 0 and φ^{ε} ≤ 0 to obtain

\begin{matrix}0 ≥ − ∫_{ℝ^{d}} φ^{ε}χ_{R}dσ_{0} = & ∫_{0}^{1} ∫_{ℝ^{d}} χ_{R} \frac{dφ^{ε}}{dt} + ⟨v_{t}, χ_{R}∇φ^{ε} + φ^{ε}∇χ_{R}⟩dσ_{t}dt \\ = & ∫_{0}^{1} ∫_{ℝ^{d}} χ_{R}(ψ + ⟨v_{t} − w_{t}^{ε}, ∇φ^{ε}⟩)dσ_{t}dt + ∫_{0}^{1} ∫_{ℝ^{d}} φ^{ε}⟨∇χ_{R}, v_{t}⟩dσ_{t}dt \\ ≥ & ∫_{0}^{1} ∫_{ℝ^{d}} χ_{R}(ψ + ⟨v_{t} − w_{t}^{ε}, ∇φ^{ε}⟩)dσ_{t}dt − ∫_{0}^{1} ∫_{ℝ^{d}} |∇χ_{R}||v_{t}|d|σ_{t}|dt. \end{matrix}

Letting ε → 0^{+} and using the uniform bound on |∇φ^{ε}| and the fact that w_{t} = v_{t} on supp χ_{R} × [0, 1], we get

∫_{0}^{1} ∫_{ℝ^{d}} χ_{R}ψdσ_{t}dt ≤ ∫_{0}^{1} ∫_{ℝ^{d}} |∇χ_{R}||v_{t}|d|σ_{t}|dt ≤ \frac{2}{R} ∫_{0}^{1} ∫_{R≤|x|≤2R} |v_{t}|d|σ_{t}|dt.

Eventually letting R → ∞ we obtain that ∫_{0}^{1} ∫_{ℝ^{d}} ψdσ_{t}dt ≤ 0. Since ψ is arbitrary we get the thesis. □

Lemma 3.3. Let μ ∈ 𝒫_{2}(ℝ^{d}) be a measure and let E be a ℝ^{m} valued measure on ℝ^{d} with finite total variation and absolutely continuous w.r.t. μ. Then for any convolution kernel ρ it holds

∫ |\frac{E ∗ ρ}{μ ∗ ρ}|^{2} μ ∗ ρdx ≤ ∫ |\frac{E}{μ} |^{2} dμ

Proof. We use Jensen inequality in the following form: if Ψ : ℝ^{m+1} → [0, +∞] is convex, l.s.c. and positively 1-homogeneous, then

Ψ (∫ ψ(x)dθ(x)) ≤ ∫ Ψ(ψ(x))dθ(x),

for any Borel map ψ : ℝ^{d} → ℝ^{m+1} and any positive and finite measure θ in ℝ^{d} (by rescaling θ to be a probability measure and looking at the image measure ψ_{#}θ the formula reduces to the standard Jensen inequality). Fix x ∈ ℝ^{d} and apply the inequality above with ψ := (E/μ, 1), θ = ρ(x − ·)μ and

Ψ(z, t) := \{ \begin{matrix}\frac{|z|^{2}}{t} & if t > 0, \\ 0 & if (z, t) = (0, 0), \\ +∞ & elsewhere, \end{matrix}

to obtain

\begin{matrix}|\frac{E ∗ ρ(x)}{μ ∗ ρ(x)}|^{2} μ ∗ ρ(x) = & Ψ (∫ \frac{E}{μ} (y)ρ(x − y)dμ(y), ∫ ρ(x − y)dμ(y)) \\ ≤ & ∫ Ψ(\frac{E}{μ} (y), 1)ρ(x − y)dμ(y) \\ = & ∫ |\frac{E}{μ} |^{2} (y)ρ(x − y)dμ(y). \end{matrix}

Integrating w.r.t. x we get the thesis. □

Theorem 3.4 (Characterization of AC curves). Let μ_{t} : [0, 1] → 𝒫_{2}(ℝ^{d}) be an absolutely continuous curve. Then for a.e. t ∈ [0, 1] there exist vector fields v_{t} ∈ L^{2}(μ_{t}, ℝ^{d}) whose norm is controlled by the metric derivative of μ_{t}

∥v_{t}∥_{μ_{t}} ≤ |μ̇_{t}|, a.e. t ∈ [0, 1],

such that the equation

(3.3)  \frac{d}{dt}μ_{t} + ∇ · (v_{t}μ_{t}) = 0,

holds in the sense of distributions, i.e.

(3.4)  ∫_{0}^{1} ∫ (\frac{d}{dt}φ(t, x) + ⟨∇_{x}φ(x, t), v_{t}(x)⟩) dμ_{t}(x)dt = 0,

for all φ ∈ C_{c}^{∞}((0, 1) × ℝ^{d}).

Conversely if μ_{t} : [0, 1] → 𝒫_{2}(ℝ^{d}) satisfies equation (3.3) for some family of vector fields v_{t} ∈ L^{2}(μ, ℝ^{d}) such that

∫_{0}^{1} ∥v_{t}∥_{μ_{t}} < +∞,

then t → μ_{t} admits an absolutely continuous representative t → \overline{μ}_{t} satisfying

| ˙\overline{μ}_{t}| ≤ ∥v_{t}∥_{\overline{μ}_{t}}, a.e. t ∈ [0, 1].

Proof. Let us start assuming that μ_{t} is an absolutely continuous curve. Using lemma 3.1 we may assume that the metric derivative is constant. For any map φ ∈ C_{c}^{1}(ℝ^{d}) consider the upper semicontinuous map H_{φ}(x, y) defined as

H_{φ}(x, y) := \{ \begin{matrix}|∇φ(x)| & if x = y, \\ \frac{|φ(x) − φ(y)|}{|x − y|} & if x ≠ y, \end{matrix}

moreover let γ_{t}^{s} be any optimal plan for the couple μ_{t}, μ_{s}. With this notation we evaluate the following limit for every t in which μ_{t} is metrically differentiable

\overline{lim}_{h→0^{+}} \frac{|μ_{t+h}(φ) − μ_{t}(φ)|}{h} = \overline{lim}_{h→0^{+}} \frac{1}{h} ∫_{ℝ^{2d}} |φ(y) − φ(x)|dγ_{t}^{t+h}(x, y)
≤ \overline{lim}_{h→0^{+}} \frac{1}{h} ∫_{ℝ^{2d}} |x − y|H_{φ}(x, y)dγ_{t}^{t+h}(x, y)

≤ \overline{lim}_{h→0^{+}} \frac{W (μ_{t+h}, μ_{t})}{h} (∫_{ℝ^{2d}} H_{φ}^{2}(x, y)dγ_{t}^{t+h}(x, y))^{1/2}
≤|μ̇_{t}|∥∇φ∥_{μ_{t}},

(3.5)

where the last inequality follows from the upper semicontinuity of H_{φ} and the fact that γ_{t}^{t+h} converge to (Id, Id)_{#}μ_{t} as h → 0. Consider now a function φ ∈ C_{c}^{1}((0, 1) × ℝ^{d}) and let μ ∈ 𝒫_{2}((0, 1) × ℝ^{d}) be the measure whose disintegration w.r.t. t is μ_{t}, then we have

\begin{matrix}∫_{(0,1)×ℝ^{d}} \frac{d}{dt}φ(t, x)dμ(t, x) = & lim_{h→0^{+}} ∫_{(0,1)×ℝ^{d}} \frac{φ(t + h, x) − φ(t, x)}{h}dμ \\ = & lim_{h→0^{+}} ∫_{(0,1)} −\frac{μ_{t}(φ(t, x)) − μ_{t−h}φ(t, x)}{h}dt, \end{matrix}

therefore using equation (3.5) and Fatou’s lemma we have

(3.6)  |∫_{(0,1)×ℝ^{d}} \frac{d}{dt}φ(t, x)dμ(t, x)| ≤ ∫_{(0,1)} |μ̇_{t}|∥∇φ(t, x)∥_{μ_{t}}dt.

Now consider the space V defined as the closure w.r.t. L_{μ}^{2} of the set {∇φ : φ ∈ C_{c}^{1}((0, 1) × ℝ^{d})} and consider the linear functional L defined on gradients of regular functions as

L(∇φ) := − ∫_{(0,1)×ℝ^{d}} \frac{d}{dt}φdμ.

Equation (3.6) ensures that L can be uniquely extended to a bounded functional on the whole V whose norm is controlled by (∫_{0}^{1} |μ̇_{t}|^{2})^{1/2} which is equal to ∫_{0}^{1} |μ̇_{t}| by our assumption on the metric derivative of t → μ_{t} (here we endow V with the L^{2} norm given by μ). Since V is an Hilbert space, the functional L may be represented by the scalar product with a certain vector v(t, x): defining v_{t}(x) := v(t, x) we have

∫_{(0,1)} ∫ \frac{d}{dt}φ(t, x)dμ_{t}(x)dt = −L(∇φ) = ∫_{(0,1)} ∫ ⟨∇φ(t, x), v_{t}(x)⟩dμ_{t}(x)dt.

To conclude we need just to show that ∥v_{t}∥_{μ_{t}} ≤ |μ̇_{t}|: we already know that ∫_{0}^{1} ∥v_{t}∥_{μ_{t}} ≤ ∫_{0}^{1} |μ̇_{t}|, to prove that ∫_{t}^{s} ∥v_{t}∥_{μ_{t}} ≤ ∫_{t}^{s} |μ̇_{t}| holds for any 0 ≤ t ≤ s ≤ 1 it is sufficient to repeat the argument restricting the curve to the interval (t, s). We will come out with a vector \overline{v}(r, x) ∈ V_{t}^{s}, where V_{t}^{s} is the closure of \{∇φ : φ ∈ C_{c}^{1}((t, s) × ℝ^{d})\} w.r.t. the L^{2} norm given by μ|_{(t,s)×ℝ^{d}}, such that ∫_{t}^{s} ∥\overline{v}(r, ·)∥_{μ_{r}}dr ≤ ∫_{t}^{s} |μ̇_{r}|dr and

∫_{(t,s)} ∫ \frac{d}{dr} φ(r, x)dμ_{r}(x)dr = ∫_{(t,s)} ∫ ⟨∇φ(r, x), \overline{v}(r, x)⟩dμ_{r}(x)dr.

Therefore it must hold \overline{v}(r, x) = v(r, x) μ−a.e. and the claim is proved.

Now we turn to the converse implication: we will assume that μ_{t} is a family of functions which satisfies equation (3.4) for a certain family of vectors v_{t}(x) ∈ L_{μ_{t}}^{2} satisfying ∫_{0}^{1} ∥v_{t}∥_{μ_{t}}dt < ∞ and we wil prove that μ_{t} is (equivalent to) an absolutely continuous curve in 𝒫_{2}(ℝ^{d}) whose metric derivative is bounded by ∥v_{t}∥_{μ_{t}}.

We will proceed by approximation: consider a family of gaussian mollifiers ρ^{ε} and define

\begin{aligned}μ_{t}^{ε} & := μ ∗ ρ^{ε}, \\ v_{t}^{ε} & := \frac{(vμ) ∗ ρ^{ε}}{μ_{t}^{ε}} . \end{aligned}

For the measures μ_{ε} it clearly holds the equation

(3.7)  \frac{d}{dt}μ_{t}^{ε} + ∇ · (v_{t}^{ε}μ_{t}^{ε}) = 0.

It is possible to prove (we skip the details, see Lemma 8.1.4 of [11] for a detailed analysis) that there exists a unique family of maps X_{t}^{ε} : ℝ^{d} → ℝ^{d} such that Ẋ_{t}^{ε}(x) = v_{t}^{ε}(X_{t}^{ε}(x)) and X_{0}^{ε}(x) = x and its just a matter of calculation to show that the measures (X_{t}^{ε})_{#}μ_{0}^{ε} satisfy equation (3.7), therefore by the uniqueness proposition 3.2 we get that μ_{t}^{ε} = (X_{t}^{ε})_{#}μ_{0}^{ε} for a.e. t. This representation allows us to estimate the distance between μ_{t}^{ε} and μ_{s}^{ε}: indeed (X_{t}^{ε}, X_{s}^{ε})_{#}μ_{0}^{ε} ∈ Adm(μ_{t}^{ε}, μ_{s}^{ε}) and therefore

W^{2}(μ_{t}^{ε}, μ_{s}^{ε}) ≤ ∫ |X_{t}^{ε} − X_{s}^{ε}|^{2}dμ_{0}^{ε} ≤ (s − t) ∫ ∫_{t}^{s} |Ẋ_{r}^{ε}|drdμ_{0}^{ε} = (s − t) ∫_{t}^{s} ∫ |v_{r}^{ε}|dμ_{r}^{ε}dr

for any 0 ≤ t ≤ s ≤ 1. Since clearly μ_{t}^{ε} W −converge to μ_{t} for every t as ε goes to 0, and since by lemma 3.3 (with E := v_{t}μ_{t}) it holds ∥v_{t}^{ε}∥_{μ_{t}^{ε}} ≤ ∥v_{t}∥_{μ_{t}}, the previous inequality implies

W^{2}(μ_{t}, μ_{s}) ≤ (s − t) ∫ ∫_{t}^{s} |v_{r}|dμ_{r}dr,

which is the thesis. □

This theorem allows to prove the well-know Benamou-Brenier formula (see also [15]) for the Wasserstein distance:

W^{2}(μ^{0}, μ^{1}) = inf ∫_{0}^{1} ∥v_{t}∥_{μ_{t}}^{2}dt,

where the infimum is taken among all the absolutely continuous curves such that μ_{0} = μ^{0}, μ_{1} = μ^{1} and among all vector fields v_{t} such that (3.1) holds. Indeed on one hand it holds

(∫_{0}^{1} ∥v_{t}∥_{μ_{t}}^{2}dt)^{1/2} ≥ ∫_{0}^{1} ∥v_{t}∥_{μ_{t}}dt ≥ L(μ_{t}) ≥ W (μ_{0}, μ_{1}),

where in the second inequality we used the second part od the theorem.

On the other hand, taking as μ_{t} any constant speed geodesic connecting μ^{0} to μ^{1} and using the first part of the theorem we get

(∫_{0}^{1} ∥v_{t}∥_{μ_{t}}^{2}dt)^{1/2} = ∫_{0}^{1} ∥v_{t}∥_{μ_{t}}dt ≤ L(μ_{t}) = W (μ_{0}, μ_{1}),

2. The regular tangent space

Let us analyze some of the consequences of theorem 3.4. Given an absolutely continuous curve t → μ_{t} the set of those families of vectors v_{t} ∈ L^{2}(μ_{t}, ℝ^{d}) satisfying (3.3) is not uniquely determined: indeed it is sufficient to choose w_{t} ∈ L^{2}(μ_{t}, ℝ^{d}) such that ∇·(w_{t}μ_{t}) = 0 and to add these vectors to the v_{t}’s to find another solution. However the following considerations suggest that there is a natural choice for the v_{t}’s.

• The second implication of theorem 3.4 shows that for any solution v_{t} it holds ∥v_{t}∥_{μ_{t}} ≥ |μ̇_{t}|.

• The first implication ensures that there exists at least one choice such that

(3.8)  ∥v_{t}∥_{μ_{t}} = |μ̇_{t}|.

• The linearity w.r.t. v_{t} of equation (3.3) and the strict convexity of the L^{2} norms give the uniqueness of those v_{t}’s which satisfy (3.8).

• Equation (3.4) ensures that the vectors v_{t} act only (through the Riesz isomor-

phism of L^{2}(μ_{t}, ℝ^{d})) on the gradient of regular functions. These facts lead us to the following definition.

Definition 3.5 (Regular tangent space). The regular tangent space (or simply tangent space) to the space 𝒫_{2}(ℝ^{d}) at a measure μ is

Tan_{μ}(𝒫_{2}(ℝ^{d})) := \overline{\{∇φ : φ ∈ C_{c}^{∞}(ℝ^{d})\}}^{L^{2}(μ,ℝ^{d})}.

Tangent vectors are characterized by the following property:

(3.9)  ∥v∥_{μ} ≤ ∥v + w∥_{μ},  ∀w ∈ L^{2}(μ, ℝ^{d}) such that ∇ · (wμ) = 0.

Indeed (3.9) holds if and only if

(3.10)  ∫ ⟨v, w⟩dμ = 0,  ∀w ∈ L^{2}(μ, ℝ^{d}) such that ∇ · (wμ) = 0,

and this is true if and only if v ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})). Equation (3.9) ensures that the two possible choices for the vectors v_{t} suggested by the previous considerations (solution of minimal norm or gradient) actually give the same vectors. We will denote by P_{μ} the orthogonal projection of L^{2}(μ, ℝ^{d}) into Tan_{μ}(𝒫_{2}(ℝ^{d})), which is obviously characterized by

(3.11)  ∫ ⟨v − P_{μ}(v), v′⟩dμ = 0 ∀v′ ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})).

Let us also introduce the normal space, i.e. the orthogonal complement of Tan_{μ}(𝒫_{2}(ℝ^{d})):

(3.12)  Tan_{μ}^{⊥}(𝒫_{2}(ℝ^{d})) := {w ∈ L^{2}(μ, ℝ^{d}) : ∇ · (wμ) = 0}.

The analogous of equation (3.9) is: w ∈ Tan_{μ}^{⊥}(𝒫_{2}(ℝ^{d})) if and only if

(3.13)  ∥v∥_{μ} ≤ ∥v + w∥_{μ},  ∀v ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})).

From now on given an absolutely continuous curve μ_{t} we will refer to the vectors v_{t} ∈ Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})) which solve (3.3) as the velocity vectors of the curve.

In order to enlight notation we will write ⟨·, ·⟩_{μ} for the scalar product in L^{2}(μ, ℝ^{d}):

⟨v, v′⟩_{μ} := ∫ ⟨v, v′⟩dμ ∀v, v′ ∈ L^{2}(μ, ℝ^{d}).

It is important to underline immediately that the structure given by the tangent spaces does not yield really an infinite dimensional Riemannian manifold. This is due to the fact that the injectivity radius of this “manifold” is 0. Before studying this problem we define the exponential map. Let us start with the following observation: if a constant speed geodesic [0, 1] ∋ t → μ_{t} is induced by an optimal map T, that is μ_{t} = (Id + t(T − Id))_{#}μ_{0}, then the continuity equation is fulfilled with

v_{t} = (T − Id) ∘ (Id + t(T − Id))^{−1},

and since ∥v_{t}∥_{μ_{t}} = ∥T − Id∥_{μ_{0}} = W (μ_{0}, μ_{1}) for any t ∈ [0, 1], those v_{t} are tangent (at least for a.e. t). Since v_{0} = T − Id we are led to the following definition:

Definition 3.6 (The exponential map). Let μ ∈ 𝒫_{2}(ℝ^{d}) be a measure. The exponential map is defined as

(3.14)  L^{2}(μ, ℝ^{d}) ∋ v → exp_{μ}(v) := (Id + v)_{#}μ.

Since the defining formula makes sense for any vector field in L^{2}(μ, ℝ^{d}) we didn’t restrict the definition to the tangent space.

Let us now come back to the question about the injectivity radius: it is clear that the map t → exp_{μ}(tv) defines a geodesic in the interval [0, \overline{t}] if and only if Id + \overline{t}v is an optimal map, and this is true if and only if I +\overline{t}Dv ≥ 0 in distributional sense (here I is the identity matrix). So t → exp_{μ}(tv) defines a geodesic in some interval if and only if Dv is bounded from below, but since Tan_{μ}(𝒫_{2}(ℝ^{d})) is defined as the L^{2}(μ, ℝ^{d}) closure of the set of gradients without taking in account any regularity, simple counterexamples show that Dv is generally unbounded.

The same argument shows that for any φ ∈ C_{c}^{∞}(ℝ^{d}) the map Id + t∇φ is optimal for sufficiently small t and therefore it holds

lim_{t→0^{+}} \frac{W (exp_{μ}(t∇φ), μ)}{t} = ∥∇φ∥_{μ}.

Now, using the inequality

W (exp_{μ}(tv), exp_{μ}(t∇φ)) ≤ t∥v − ∇φ∥_{μ},

it is easy to check that for every v ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) it holds

lim_{t→0^{+}} \frac{W (exp_{μ}(tv), μ)}{t} = ∥v∥_{μ},

i.e. the infinitesimal behavior of the exponential map on the tangent space is the right one. We will show in the next chapter (see theorem 4.19) that the previous formula characterizes the tangent vectors.

The next proposition proves an important property of velocity vectors: they can be recovered as limit of optimal transport plans (or maps, if any). It is important to underline that even if there is no hypothesis on the regularity of the measures μ_{t}, the limit plan is induced by the map v_{t}. It is for this reason that in the studies of differential properties along absolutely continuous curves it suffices to consider the regular tangent space.

Proposition 3.7. Let μ_{t} be an absolutely continuous curve and let v_{t} be its velocity vectors. Then for almost every t ∈ [0, 1] it holds: for any choice of γ_{t}^{t+h} ∈ Opt(μ_{t}, μ_{t+h}) we have

(3.15)  lim_{h→0} (π^{1}, \frac{π^{2} − π^{1}}{h} )_{#}γ_{t}^{t+h} → (Id, v_{t})_{#}μ_{t},

and

(3.16)  lim_{h→0} \frac{W (μ_{t+h}, (Id + hv_{t})_{#}μ_{t})}{h} = 0.

In particular for almost every t ∈ [0, 1] such that μ_{t} ∈ 𝒫_{2}^{r}(ℝ^{d}) it holds

(3.17)  \frac{T_{μ_{t}}^{μ_{t+h}} − Id}{h} → v_{t}, in L^{2}(μ_{t}, ℝ^{d}).

Proof. Theorem 3.4 ensures that it is sufficient to prove (3.15) and (3.16) for every t ∈ [0, 1] such that

(3.18)  \frac{W (μ_{t+h}, μ_{t})}{h} → |μ̇_{t}| = ∥v_{t}∥_{μ_{t}}.

So, fix t satisfying (3.18), define

η_{h} := (π^{1}, \frac{π^{2} − π^{1}}{h} )_{#}γ_{t}^{t+h},

and observe that the family η_{h} is bounded. Letting η be any τ accumulation point of η_{h} as h → 0, the inequality

∥ℬ(η)∥_{μ_{t}} ≤ ∥x_{2}∥_{η} ≤ \underline{lim}_{h→0} ∥x_{2}∥_{η_{h}} = lim \frac{W (μ_{t+h}, μ_{t})}{h} = ∥v_{t}∥_{μ_{t}},

and equations (3.9), (2.31) ensure that in order to prove (3.15) it suffices to show that

(3.19)  ∇ · ((ℬ(η) − v_{t})μ_{t}) = 0.

Now fix φ ∈ C_{c}^{∞}(ℝ^{d}), observe that

\begin{matrix}\frac{μ_{t+h}(φ) − μ_{t}(φ)}{h} = & \frac{1}{h} ∫ φ(x_{2}) − φ(x_{1})dγ_{t}^{t+h} \\ = & \frac{1}{h} ∫ φ(x_{1} + hx_{2}) − φ(x_{1})dη_{h} = ∫ ⟨∇φ(x_{1}), x_{2}⟩dη_{h} + R_{h}, \end{matrix}

where |R_{h}| ≤ h Lip_{∇φ} ∥x_{2}∥_{η_{h}}, and let h tend to 0 to obtain

∫ ⟨∇φ, v_{t}⟩dμ_{t} = ∫ ⟨∇φ(x_{1}), x_{2}⟩dη = ∫ ⟨∇φ, ℬ(η)⟩dμ_{t}.

The arbitrariness of φ gives (3.19).

Now we turn to (3.16). Observe that since

(π^{1} + hv_{t} ∘ π^{1}, π^{2})_{#}γ_{t}^{t+h} ∈ Adm((Id + hv)μ_{t}, μ_{t+h})

it holds

\frac{W^{2}(μ_{t+h}, (Id + hv_{t})_{#}μ_{t})}{h^{2}} ≤ \frac{1}{h^{2}} ∥x_{1} + hv_{t}(x_{1}) − x_{2}∥_{γ_{t}^{t+h}}^{2} = ∥v_{t}(x_{1}) − x_{2}∥_{η_{h}}^{2}.

Now observe that for any φ ∈ C_{c}^{∞}, equation (3.15) and the fact that the function (x_{1}, x_{2}) → (∇φ(x_{1}) − x_{2}) has linear growth, give ∥∇φ(x_{1}) − x_{2}∥_{η_{h}} → ∥∇φ(x_{1}) − v_{t}(x_{1})∥_{μ_{t}}. Therefore we have

lim_{h→0} \frac{W (μ_{t+h}, (Id + hv_{t})_{#}μ_{t})}{h} ≤ lim_{h→0} ∥v_{t}(x_{1}) − x_{2}∥_{η_{h}}

≤ ∥v_{t} − ∇φ∥_{μ_{t}} + lim_{h→0} ∥∇φ(x_{1}) − x_{2}∥_{η_{h}}

= 2∥v_{t} − ∇φ∥_{μ_{t}}.

Letting ∇φ tend to v_{t} in L_{μ_{t}}^{2} we get the thesis.

The last statement is a consequence of (3.15) and proposition 2.23. □

As a first application of the definition of tangent space we discuss the differentiability properties of the functional W^{2}(·, σ).

Proposition 3.8 (Differentiability of W^{2} at regular measures). Let σ ∈ 𝒫_{2}(ℝ^{d}) be a fixed measure. Then the functional

𝒫_{2}(ℝ^{d}) ∋ μ → W^{2}(μ, σ),

is differentiable at any regular measure μ and its differential is 2(Id − T_{μ}^{σ}), i.e. the following equality holds for any μ ∈ 𝒫_{2}^{r}(ℝ^{d}):

(3.20)  lim_{ν→^{W}μ} \frac{W^{2}(ν, σ) − W^{2}(μ, σ) − ⟨2(Id − T_{μ}^{σ}), T_{μ}^{ν} − Id⟩_{μ}}{W (ν, μ)} = 0.

Proof. The \overline{lim} inequality follows by

W^{2}(ν, σ) ≤ ∥T_{μ}^{ν} − T_{μ}^{σ}∥_{μ}^{2} = W^{2}(μ, σ) + W^{2}(μ, ν) + 2⟨T_{μ}^{ν} − Id, Id − T_{μ}^{σ}⟩_{μ}.

To get the other one, fix any sequence ν_{n} → μ and choose α_{n} such that

\begin{aligned}π_{#}^{1,2}α_{n} & = (Id, T_{μ}^{ν_{n}})_{#}μ, \\ π_{#}^{2,3}α_{n} & ∈ Opt(ν_{n}, σ), \end{aligned}

to obtain

W^{2}(μ, σ) ≤ ∥x_{1} − x_{3}∥_{α_{n}}^{2}

= W^{2}(μ, ν_{n}) + W^{2}(ν_{n}, σ) + 2 ∫ ⟨x_{1} − T_{μ}^{ν_{n}}(x_{1}), T_{μ}^{ν_{n}}(x_{1}) − x_{3}⟩dγ_{n}(x_{1}, x_{3}),

where γ_{n} := π_{#}^{1,3}α_{n}.

Conclude observing that since ∥x_{2} −x_{3}∥_{γ_{n}} ≤ W (μ, ν_{n})+W (ν_{n}, σ), proposition 2.19

implies γ_{n} →^{W} (Id, T_{μ}^{σ})_{#}μ and so it holds

| ∫ ⟨x_{1}−T_{μ}^{ν_{n}}(x_{1}), T_{μ}^{ν_{n}}(x_{1}) − x_{3}⟩dγ_{n}(x_{1}, x_{3}) − ∫ ⟨x_{1} − T_{μ}^{ν_{n}}(x_{1}), x_{1} − T_{μ}^{σ}(x_{1})⟩dμ(x_{1})|

= |∫ ⟨x_{1} − T_{μ}^{ν_{n}}(x_{1}), T_{μ}^{ν_{n}}(x_{1}) − x_{3} − x_{1} + T_{μ}^{σ}(x_{1})⟩dγ_{n}|
≤ W (μ, ν_{n})∥T_{μ}^{ν_{n}}(x_{1}) − x_{3} − x_{1} + T_{μ}^{σ}(x_{1})∥_{γ_{n}} = o(W (μ, ν_{n})).
□

It is interesting to notice the following formal analogy with the usual Riemannian case: if M is a Riemannian manifold, the differential of the function x → d^{2}(x, z), provided it exists, is −2 exp_{x}^{−1}(z). This means that it holds

lim_{y→x} \frac{d^{2}(y, z) − d^{2}(x, z) − ⟨ − 2 exp_{x}^{−1}(z), exp_{x}^{−1}(y)⟩_{x}}{d(x, y)} = 0,

where ⟨·, ·⟩_{x} is the scalar product on T_{x}M.

Now observe that in the space (𝒫_{2}(ℝ^{d}), W), for a given μ ∈ 𝒫_{2}^{r}(ℝ^{d}), the natural right inverse of v → exp_{μ}(v) = (Id + v)_{#}μ is ν → exp_{μ}^{−1}(ν) := T_{μ}^{ν} − Id. This means that T_{μ}^{ν} − Id is the only map f ∈ L_{μ}^{2} such that [0, 1] ∋ t → exp_{μ}(tf) is a geodesic connecting μ to ν. Therefore equation (3.20) is the perfect analogous of the formula valid in the usual Riemannian setting. We will investigate in the next Chapter how to generalize the concept of exp_{μ}^{−1} to non-regular measures, and in Chapter 5 the general differentiability properties of μ → W^{2}(μ, σ) (see theorem 5.20 and corollary 5.22).

A simple consequence of the previous proposition and of (3.15) is the following formula for the derivative of W^{2}(·, σ) along absolutely continuous curves with range on regular measures:

\frac{d}{dt}W^{2}(μ_{t}, σ) = 2⟨v_{t}, Id − T_{μ_{t}}^{σ}⟩_{μ_{t}}, a.e. t ∈ [0, 1]

Since, as we will see in corollary 5.22, W^{2} is generally not differentiable on non regular measures, it is quite surprising that a similar formula holds even without any assumption on μ_{t}:

Proposition 3.9. Let μ_{t} : [0, 1] → 𝒫_{2}(ℝ^{d}) be an absolutely continuous curve, v_{t} its velocity vector field and σ ∈ 𝒫_{2}(ℝ^{d}) a fixed measure. Then for a.e. t it holds:

(3.21)  \frac{d}{dt}W^{2}(μ_{t}, σ) = 2 ∫ ⟨v_{t}(x_{1}), x_{1} − x_{2}⟩dγ(x_{1}, x_{2}), ∀γ ∈ Opt(μ_{t}, σ).

Proof. Observe that the map t → W (μ_{t}, σ) is absolutely continuous, so the same is true for t → W^{2}(μ_{t}, σ) and the left hand side of (3.21) exists for a.e. t. We will show that claimed property is true for every t such that (3.16) holds and the derivative of W (μ_{t}, σ) exists; fix such a t and observe that equation (3.16) ensures that

\frac{d}{ds}W^{2}(μ_{s}, σ)|_{s=t} = lim_{h→0} \frac{W^{2}((Id + hv_{t})_{#}μ_{t}, σ) − W^{2}(μ_{t}, σ)}{h}.

Now choose any γ ∈ Opt(μ_{t}, σ) and note that since (π^{1} + hv_{t} ∘ π^{1}, π^{2})_{#}γ ∈ Adm((Id + hv_{t})_{#}μ_{t}, σ) it holds

\begin{matrix}W^{2}((Id + hv_{t})_{#}μ_{t}, σ) ≤ & ∥x_{1} + hv_{t}(x_{1}) − x_{2}∥_{γ}^{2} \\ = & W^{2}(μ_{t}, σ) + h^{2}∥v_{t}∥_{μ_{t}}^{2} + 2h ∫ ⟨v_{t}(x_{1}), x_{1} − x_{2}⟩dγ \\ = & W^{2}(μ_{t}, σ) + h^{2}∥v_{t}∥_{μ_{t}}^{2} − 2h⟨v_{t}, ℬ(γ) − Id⟩_{μ_{t}}, \end{matrix}

and the thesis follows dividing by h and letting h ↑ 0 and h ↓ 0. □

Observe that the equation (3.21) may be written as

\frac{d}{dt}W^{2}(μ_{t}, σ) = 2⟨v_{t}, Id − ℬ(γ)⟩_{μ_{t}}.

We will show in theorem 4.15 that the barycentric projection of an optimal plan belongs to the regular tangent space, and usually different optimal plans have different barycentric projections: so it is a non trivial fact that almost everywhere along an absolutely continuous curve the barycentric projections of optimal plans act in the same way on the velocity vector field.

CHAPTER 4 The geometric tangent space

1. An introduction to Tan_{μ}(𝒫_{2}(ℝ^{d}))

The regular tangent space is a very useful tool in the study of differentiability properties along absolutely continuous curves or on regular measures, however the study of the properties of geodesically convex functionals (see the next chapter) shows that in order to reproduce in the framework of the Wasserstein distance the classical results of convex analysis, an enlargement of Tan_{μ}(𝒫_{2}(ℝ^{d})) is necessary. Even from a theoretical point of view the definition of Tan_{μ}(𝒫_{2}(ℝ^{d})) is not completely satisfactory: for instance if μ = δ_{x} is a delta then Tan_{μ}(𝒫_{2}(ℝ^{d})) is isomorphic to ℝ^{d} (and so it is very small) and the range of the exponential map is just the set of all δ’s. This is due, of course, to the fact that working with maps does not allow to split masses. In this section we will describe another tangent space which we call Tan_{μ}(𝒫_{2}(ℝ^{d})), made with plans, which will be a subset of

(4.1)  𝒫_{2}(ℝ^{2d})_{μ} := {γ ∈ 𝒫_{2}(ℝ^{2d}) : π_{#}^{1}γ = μ},

and we will show that for any measure μ the map

(4.2)  \begin{aligned}ℑ_{μ} : Tan_{μ}(𝒫_{2}(ℝ^{d})) & → & 𝒫_{2}(ℝ^{2d})_{μ} \\ v & → & (Id, v)_{#}μ, \end{aligned}

will be an embedding of Tan_{μ}(𝒫_{2}(ℝ^{d})) into Tan_{μ}(𝒫_{2}(ℝ^{d})), and that this embedding is actually an isomorphism if the base measure μ is regular.

Before going on we want to underline that this section is the most technical part of the thesis. Working with plans is not very intuitive, and many proofs will need the introduction of multiple plans of dimension 4 or even 5. However the structure of 𝒫_{2}(ℝ^{2d})_{μ} is formally very similar to the one of Hilbert spaces: hoping for a better understanding we will sometimes recall the proof of some basic theorems in Hilbert setting before proving, with the same ideas, their analogous in 𝒫_{2}(ℝ^{2d})_{μ}.

To understand the heuristic idea behind the definition of the geometric tangent space let us think for a moment to the case of a Riemannian manifold M embedded in some ℝ^{d}. The tangent space at point x ∈ M may be defined in the following way: for any geodesic t → x(t) starting from x, i.e. x(0) = x, we try to find the limit in ℝ^{d} of

(4.3)  \frac{x(t) − x(0)}{t},

and, whenever this limit exists, we put it into the tangent space, which therefore will be the set of all possible limits of the previous kind.

Trying to do the same in the Wasserstein framework arise the following problems: what is the natural analogous for equation (4.3) in our setting? And: what is the topology we use to calculate the limit?

For regular base measure μ the answer to the first question should be

(4.4)  \frac{T_{μ}^{μ_{t}} − Id}{t},

where t → μ_{t} is a given geodesic starting from μ parametrised in some right neighborhood of 0. Remark 2.10, and in particular (2.12), ensures that for some optimal map T and \overline{t} > 0 it holds

\frac{T_{μ}^{μ_{t}} − Id}{t} = \frac{T − Id}{\overline{t}}, for t < \overline{t}.

In particular there is no problem in finding the limit and we can affirm that the geometric tangent space at a regular measure μ should contain at least all the maps of the form

v := \frac{T − Id}{\overline{t}},

where T varies among all possible optimal maps and \overline{t} among positive reals. Now observe that since all the maps belong to L^{2}(μ, ℝ^{d}) it is natural to equip the tangent space with the L^{2} distance w.r.t. μ. Since we want to deal with a complete tangent space we define

(4.5) Tan_{μ}(𝒫_{2}(ℝ^{d})):=\overline{\{v ∈ L^{2}(μ, ℝ^{d}) : Id + tv is optimal for some t > 0\}}^{L^{2}(μ,ℝ^{d})}

Now let us come back to the general case, i.e. μ ∈ 𝒫_{2}(ℝ^{d}) is not necessarily regular. Let t → μ_{t} be a constant speed geodesic starting from μ and parametrised in some right neighborhood of 0. Keeping in mind the correspondence

T → (Id, T)_{#}μ,

from maps to plans, and equation (4.4), we are led to try to find the limit of

(4.6)  (π^{1}, \frac{π^{2} − π^{1}}{t} )_{#} γ_{t},  γ_{t} ∈ Opt(μ, μ_{t}).

Using corollary 2.9 we know that there exist an optimal transport plan \overline{γ} and a positive \overline{t} such that for t < \overline{t} it holds

γ_{t} = (π^{1}, π^{1} + \frac{t}{\overline{t}}(π^{2} − π^{1}))_{#} \overline{γ},

from which we get that the expression in (4.6) is once again independent on small t and equal to

γ := (π^{1}, \frac{1}{\overline{t}}(π^{2} − π^{1}))_{#} \overline{γ}.

Inverting this equation we define the following set:

GEOD_{μ} := {γ ∈ 𝒫_{2}(ℝ^{2d})_{μ} : (π^{1}, π^{1} + tπ^{2})_{#}γ is optimal for some t > 0}.

1. AN INTRODUCTION TO Tan_{μ}(𝒫_{2}(ℝ^{d})) 43

Analogously to the previous case we have

GEOD_{μ} ⊂ 𝒫_{2}(ℝ^{2d})_{μ},

and we want to endow 𝒫_{2}(ℝ^{2d})_{μ} with some distance and then define the geometric tangent space as the closure of GEOD_{μ} w.r.t. this distance. Since we want the map ℑ_{μ} defined in (4.2) to be an isometry, we define, for any couple γ_{1}, γ_{2} ∈ 𝒫_{2}(ℝ^{2d})_{μ}, the distance W_{μ} (γ_{1}, γ_{2}) as

(4.7)  W_{μ}^{2} (γ_{1}, γ_{2}) := ∫ W^{2}((γ_{1})_{x}, (γ_{2})_{x})dμ(x),

where (γ_{1})_{x}, (γ_{2})_{x} are the disintegrations of γ_{1}, γ_{2} w.r.t. μ.

Finally we give the following definition:

Definition 4.1 (The geometric tangent space). Let μ ∈ 𝒫_{2}(ℝ^{d}) be a measure. The geometric tangent space Tan_{μ}(𝒫_{2}(ℝ^{d})) of 𝒫_{2}(ℝ^{d}) at μ is the closure of GEOD_{μ} in 𝒫_{2}(ℝ^{2d})_{μ}with respect to the distance W_{μ}.

Actually we have still to show that W_{μ} is a distance and that the geometric tangent space is complete (this is not automatic from the definition since we have to prove at first that (𝒫_{2}(ℝ^{2d})_{μ}, W_{μ}) is complete). These are the goals of the foregoing discussion.

From now on μ ∈ 𝒫_{2}(ℝ^{d}) is a fixed measure, and its variable will always be x_{0}. A generic plan γ ∈ 𝒫_{2}(ℝ^{2d})_{μ} will therefore belong to 𝒫_{2}(ℝ_{0}^{d} × ℝ_{i}^{d}) for some i = 1, 2, . . .. For a given couple γ_{1}, γ_{2} the set of admissible 3-plan is defined as

ADM_{μ}(γ_{1}, γ_{2}) := {α ∈ 𝒫_{2}(ℝ_{0}^{d} × ℝ_{1}^{d} × ℝ_{2}^{d}) : π_{#}^{0,1}α = γ_{1}, π_{#}^{0,2}α = γ_{2}}.

The cost associated to a 3-plan α is

∥x_{1} − x_{2}∥_{α} = (∫ |x_{1} − x_{2}|^{2}dα)^{1/2} ,

and the proposition below justifies this definition.

Proposition 4.2. Let γ_{1}, γ_{2} ∈ 𝒫_{2}(ℝ^{2d})_{μ} be any couple of plans. Then
(4.8)  W_{μ} (γ_{1}, γ_{2}) = min_{α∈ADM_{μ}(γ_{1},γ_{2})} ∥x_{1} − x_{2}∥_{α}.

Proof. Let α ∈ ADM_{μ}(γ_{1}, γ_{2}) be any admissible 3-plan and dα = dμ × dα_{x_{0}} be its disintegration w.r.t. μ. Since for μ-a.e. x_{0} it holds α_{x_{0}} ∈ Adm((γ_{1})_{x_{0}}, (γ_{2})_{x_{0}}) we have

\begin{matrix}∥x_{1} − x_{2}∥_{α}^{2} = & ∫ ∥x_{1} − x_{2}∥^{2}dα = ∫ (∥x_{1} − x_{2}∥^{2}dα_{x_{0}})dμ(x_{0}) \\ ≥ & ∫ W^{2}((γ_{1})_{x_{0}}, (γ_{2})_{x_{0}})dμ(x_{0}) = W_{μ}^{2} (γ_{1}, γ_{2}) . \end{matrix}

To obtain the other inequality use the approximation lemma below on the plans γ_{1}, γ_{2} and observe that for every n ∈ ℕ the set Opt((γ_{1}^{n})_{x_{0}}, (γ_{2}^{n})_{x_{0}},) contains only one element α_{x_{0}}^{n} and that, by proposition 2.19 the function x_{0} → α_{x_{0}}^{n} is continuous from ℝ^{d} to 𝒫_{2}(ℝ^{d}) endowed with the weak topology given by the duality with C_{c} functions. Therefore there exists a 3-plan α^{n} whose disintegration w.r.t. x_{0} is α_{x_{0}}^{n}, and it holds

∥x_{1} − x_{2}∥_{α^{n}}^{2} = ∫ ∥x_{1} − x_{2}∥_{α_{x_{0}}^{n}}^{2} dμ = ∫ W^{2}((γ_{1}^{n})_{x_{0}}, (γ_{2}^{n})_{x_{0}})dμ = W^{2}(γ_{1}^{n}, γ_{2}^{n}).

Letting n tend to ∞ the proof is achieved. □

Lemma 4.3. Let γ ∈ 𝒫_{2}(ℝ^{2d})_{μ}. Then there exists a sequence of plans (γ^{n}) ⊂ 𝒫_{2}(ℝ^{2d})_{μ} such that

i) (γ^{n})_{x_{0}} is regular for μ − a.e. x_{0},

ii) x → (γ^{n})_{x_{0}} is continuous as a function from ℝ^{d} to (𝒫_{2}(ℝ^{d}), W),
iii) ∫ W^{2}((γ^{n})_{x_{0}}, γ_{x_{0}})dμ(x_{0}) → 0.

Proof. Let X_{N} ⊂ 𝒫_{2}(ℝ^{d}) be the set of measures whose second moment is bounded by N endowed with the distance W (which induces the weak topology given by the duality with C_{c} functions). Define Y_{N} ⊂ 𝒫_{2}(ℝ^{2d})_{μ} to be the set of those plans \overline{γ} such that (\overline{γ})_{x_{0}} ∈ X_{N} for μ−a.e. x_{0}, and assume for a moment that γ ∈ Y_{N} for a certain N. Then we know that the Borel function from ℝ^{d} to X_{N} given by x_{0} → γ_{x_{0}} may be approximated μ−a.e. with a sequence of continuous functions x_{0} → σ_{x_{0}}^{n} ∈ X_{N}. Now let (ρ^{n}) be any sequence of mollifiers on ℝ^{d} and define γ_{x_{0}}^{n} := σ_{x_{0}}^{n} ∗ ρ^{n}. It is clear that γ_{x_{0}}^{n} ∈ X_{N′} for some N′ ≥ N independent on n, and that x_{0} → γ_{x_{0}}^{n} is continuous. We claim that the plans γ^{n} satisfy the thesis. We have only to show that ∫ W^{2}((γ^{n})_{x_{0}}, γ_{x_{0}})dμ(x_{0}) tends to 0. Observe that

(∫ W^{2}((γ^{n})_{x_{0}}, γ_{x_{0}})dμ(x_{0}))^{1/2}

≤ (∫ W^{2}((γ^{n})_{x_{0}}, σ_{x_{0}}^{n})dμ(x_{0}))^{1/2} + (∫ W^{2}((σ^{n})_{x_{0}}, γ_{x_{0}})dμ(x_{0}))^{1/2} ,

and that the first integrand is bounded by ∫ |x|^{2}ρ^{n}(x)dx for every x_{0}, while the second one converges to 0 for μ−a.e. x_{0} (by construction of σ_{x_{0}}^{n}) and is bounded by 4N^{2}. The dominated convergence theorem gives the thesis.

For the general case it is sufficent to approximate at first γ with plans \overline{γ}^{n} ∈ Y_{n} and to proceed with a standard diagonalization argument. □

The previous proposition shows that the infimum is always attained. We will denote by OPT_{μ}(γ_{1}, γ_{2}) the set of those 3-plans which realize the minimum, i.e.:

(4.9)  α ∈ OPT_{μ}(γ_{1}, γ_{2}) ⟺ W_{μ} (γ_{1}, γ_{2}) = ∥x_{1} − x_{2}∥_{α}.

The proof of the previous proposition characterizes optimal 3-plans as follows

α ∈ OPT_{μ}(γ_{1}, γ_{2}) ⟺ α_{x_{0}} ∈ Opt((γ_{1})_{x_{0}}, (γ_{2})_{x_{0}}), μ-a.e. x_{0}.
Theorem 4.4. The function W_{μ} is a distance on 𝒫_{2}(ℝ^{2d})_{μ}.

Proof. Choosing α := (π^{0}, π^{1}, π^{1})_{#}γ it follows that W_{μ}(γ, γ) = 0. Conversely suppose that W_{μ}(γ_{1}, γ_{2}) = 0 and choose an optimal 3-plan α. The equality ∥x_{1} − x_{2}∥_{α} = 0 gives that for α-a.e. triple (x_{0}, x_{1}, x_{2}) it holds x_{1} = x_{2}. In particular the two functions π^{0,1}, π^{0,2} are equal in L^{2}(α, ℝ^{d}) and therefore γ_{1} = π_{#}^{0,1}α = π_{#}^{0,2}α = γ_{2}.

The equality W_{μ}(γ_{1}, γ_{2}) = W_{μ}(γ_{2}, γ_{1}) is clear.

To get the triangle inequality argue as in the proof of triangle inequality for W: given γ_{1}, γ_{2}, γ_{3} ∈ 𝒫_{2}(ℝ^{2d})_{μ} choose α_{1,2} ∈ OPT_{μ}(γ_{1}, γ_{2}), α_{2,3} ∈ OPT_{μ}(γ_{2}, γ_{3}) and find a 4-plan β satisfying

π_{#}^{0,1,2}β = α_{1,2},
π_{#}^{0,2,3}β = α_{2,3}.

Since π_{#}^{0,1,3}β ∈ ADM_{μ}(γ_{1}, γ_{3}) we have

W_{μ}(γ_{1}, γ_{3}) ≤ ∥x_{1} − x_{3}∥_{β} ≤ ∥x_{1} − x_{2}∥_{β} + ∥x_{2} − x_{3}∥_{β} = W_{μ}(γ_{1}, γ_{2}) + W_{μ}(γ_{2}, γ_{3}). Finally since W_{μ}(γ, (Id, 0)_{#}μ) = ∥x_{1}∥_{γ} < ∞, the triangular inequality ensures that W_{μ}(γ_{1}, γ_{2}) < +∞ for every couple γ_{1}, γ_{2} ∈ 𝒫_{2}(ℝ^{2d})_{μ}. □

Theorem 4.5 (l.s.c. of W_{μ} and completeness of 𝒫_{2}(ℝ^{2d})_{μ}). Let μ ∈ 𝒫_{2}(ℝ^{d}) be a measure. Then the distance W_{μ} (·, ·) is lower semicontinuous with respect to the weak convergence in 𝒫_{2}(ℝ^{2d}) and the space 𝒫_{2}(ℝ^{2d})_{μ} is complete with respect to the distance W_{μ} (·, ·).

Proof. Let (γ_{n}^{1}), (γ_{n}^{2}) ∈ 𝒫_{2}(ℝ^{2d})_{μ} be two sequences weakly converging to γ^{1}, γ^{2} in 𝒫_{2}(ℝ^{2d}) respectively and let α_{n} be optimal 3-plans between γ_{n}^{1} and γ_{n}^{2}. It is easy to see that the sequence α_{n} is tight in ℝ^{3d} and so there exists a subsequence (α_{n_{k}}) weakly converging in 𝒫_{2}(ℝ^{3d}) to a certain 3-plan α and such that lim_{k} ∥x_{1} − x_{2}∥_{α_{n_{k}}} = \underline{lim}_{n} ∥x_{1} − x_{2}∥_{α_{n}}. Given that the limit 3-plan α satisfies π_{#}^{0,1}α = γ^{1}, π_{#}^{0,2}α = γ^{2}, we have:

W_{μ} (γ^{1}, γ^{2}) ≤ ∥x_{1}−x_{2}∥_{α} ≤ lim_{k→∞} ∥x_{1}−x_{2}∥_{α_{n_{k}}} = \underline{lim}_{n→∞} ∥x_{1}−x_{2}∥_{α_{n}} = \underline{lim}_{n→∞} W_{μ} (γ_{n}^{1}, γ_{n}^{2}) .

Now we turn to the completeness: assume that (γ_{n}) is a Cauchy sequence. Being this sequence bounded in 𝒫_{2}(ℝ^{2d})_{μ} it is also tight and so there exists a subsequence (γ_{n_{k}}) weakly converging to a plan γ. The lower semicontinuity yields

\overline{lim}_{m→∞} W_{μ} (γ_{m}, γ) ≤ \overline{lim}_{m→∞} \underline{lim}_{k→∞} W_{μ} (γ_{m}, γ_{n_{k}}) = 0.

□

Corollary 4.6. Tan_{μ}(𝒫_{2}(ℝ^{d})) is a complete metric space.

It should be noted carefully that the topology induced by the distance W_{μ} is not the one induced by W on 𝒫_{2}(ℝ^{2d}). For instance let d = 1, μ := L^{1}|_{[0,1]} and v_{n} := sin(nx) ∈ L^{2}(μ, ℝ). Then it is not difficult to show that n → ℑ_{μ}(v_{n}) does not have any converging subsequence in (𝒫_{2}(ℝ^{2d})_{μ}, W_{μ}). On the other hand some calculations show that ℑ_{μ}(v_{n}) tends to (1 − y^{2})^{−1/2} · L^{2}|_{[0,1]^{2}} in (𝒫_{2}(ℝ^{2d})_{μ}, W).

There is a special case in which the two topologies give the same converging sequences: when the limit plan is induced by a map.

Proposition 4.7. Let γ_{n}, γ ∈ 𝒫_{2}(ℝ^{2d})_{μ} be plans, and assume that γ s induced by a map. Then γ_{n} W_{μ}−converge to γ if and only if γ_{n} W −converge to γ.

Proof. Since W_{μ} ≥ W one implication is obvious. For the converse one, take a sequence of continuous functions f_{n} with bounded support such that f_{n} → f in L_{μ}^{2}, where f is the function which induces γ. Observe the functions |y −f_{k}(x)|^{2} are continuous on ℝ^{2d}, bounded from below and with quadratic growth, therefore propositions 2.5, 1.4 ensure that

lim_{n→∞} W_{μ}(γ_{n}, ℑ(f_{k})) = lim_{n→∞} ∥y − f_{k}(x)∥_{γ_{n}} = ∥y − f_{k}(x)∥_{γ} = ∥f − f_{k}∥_{μ}.

Therefore it holds

\overline{lim}_{n→∞} W_{μ}(γ_{n}, γ) ≤ \overline{lim}_{k→∞} \overline{lim}_{n→∞} (W_{μ}(γ_{n}, ℑ(f_{k})) + W_{μ}(ℑ(f_{k}), γ)) ≤ 2 \overline{lim}_{k→∞} ∥f − f_{k}∥_{μ} = 0.

□

2. Directional derivative of W^{2}(·, ν)

In this section we describe the first differentiability properties of W^{2}(·, ν) along curves of the kind [γ](t), where γ is any plan in 𝒫_{2}(ℝ_{1}^{d} × ℝ_{2}^{d}), not necessarily optimal.

Let us first observe that using lemma 2.12 and reasoning as in the proof of 2.13 it is possible to prove an inequality like (2.16) without any assumption on γ:

(4.10) W^{2}([γ(t)], ν) ≥ (1 − t)W^{2}([γ](0), ν) + tW^{2}(γ[1], ν) − t(1 − t)∥x_{1} − x_{2}∥_{γ}^{2}.

It is just a matter of calculations to show that this equation implies that the function

t → W^{2}([γ](t), ν) − t^{2}∥x_{1} − x_{2}∥_{γ}

is concave. Therefore from the general theory of concave function we know that for every t ∈ [0, 1) there exists the right derivative

\frac{d}{dt_{+}} W^{2}([γ](t), ν) := lim_{h→0^{+}} \frac{W^{2}([γ](t + h), ν) − W^{2}([γ](t), ν)}{h},

and, for t ∈ (0, 1], the left derivative

\frac{d}{dt_{−}} W^{2}([γ](t), ν) := lim_{h→0^{−}} \frac{W^{2}([γ](t + h), ν) − W^{2}([γ](t), ν)}{h},

and that

\frac{d}{dt_{+}} W^{2}([γ](t), ν) ≤ \frac{d}{dt_{−}} W^{2}([γ](t), ν), ∀t ∈ (0, 1).

Moreover for every t not belonging to the set N of points of non differentiability (which is at most countable), it holds

\frac{d}{dt_{+}} W^{2}([γ](t), ν) = \frac{d}{dt_{−}} W^{2}([γ](t), ν).

2. DIRECTIONAL DERIVATIVE OF W^{2}(·, ν) 47

Lemma 4.8. Let γ ∈ 𝒫_{2}(ℝ_{1}^{d} × ℝ_{2}^{d}) be a plan and let ν be a fixed measure. Then for every 3-plan μ_{t} satisfying

(4.11a)  π_{#}^{1,2}μ_{t} = γ,
(4.11b)  ((1 − t)π^{1} + tπ^{2}, π^{3})_{#}μ_{t} ∈ Opt([γ](t), ν),

it holds

\begin{matrix}\frac{d}{dt_{+}} W^{2}([γ](t), ν) ≤ & ∥x_{2} − x_{3}∥_{μ_{t}}^{2} − ∥x_{1} − x_{3}∥_{μ_{t}}^{2} + (2t − 1)∥x_{1} − x_{2}∥_{μ_{t}}^{2} \\ ≤ & \frac{d}{dt_{−}} W^{2}([γ](t), ν). \end{matrix} ; (4.12)

In particular both inequalities are equalities if t belongs to the set of differentiability of the function.

Proof. Use lemma 2.12 to find, for every t ∈ [0, 1], a 3-plan μ_{t} satisfying

\begin{aligned}π_{#}^{1,2}μ_{t} & = γ, \\ ((1 − t)π^{1} + tπ^{2}, π^{3})_{#}μ_{t} & ∈ Opt([γ](t), ν). \end{aligned}

Now fix t_{0} and take derivatives on

∥(1 − t)x_{1} + tx_{2} − x_{3}∥_{μ_{t_{0}}}^{2} = (1 − t)∥x_{1} − x_{3}∥_{μ_{t_{0}}}^{2} + t∥x_{2} − x_{3}∥_{μ_{t_{0}}}^{2} − t(1 − t)∥x_{1} − x_{2}∥_{μ_{t_{0}}}^{2} ,

to get

\frac{d}{dt}∥(1 − t)x_{1} + tx_{2} − x_{3}∥_{μ_{t_{0}}}^{2} = ∥x_{2} − x_{3}∥_{μ_{t_{0}}}^{2} − ∥x_{1} − x_{3}∥_{μ_{t_{0}}}^{2} + (2t − 1)∥x_{1} − x_{2}∥_{μ_{t_{0}}}^{2} .

To conclude it is sufficient to observe that

W^{2}([γ](t), ν) ≤ ∥(1 − t)x_{1} + tx_{2} − x_{3}∥_{μ_{t_{0}}}^{2} ,
W^{2}([γ](t_{0}), ν) = ∥(1 − t_{0})x_{1} + t_{0}x_{2} − x_{3}∥_{μ_{t_{0}}}^{2} .
□

From this lemma it is possible to derive a precise formula for the directional derivative. Let us first introduce the following set:

Definition 4.9. Given a plan γ ∈ 𝒫_{2}(ℝ_{1}^{d} × ℝ_{2}^{d}) and measure ν ∈ 𝒫_{2}(ℝ_{3}^{d}), the set Opt(γ, ν) is the set of 3-plans μ satisfying

π_{#}^{1,2}μ = γ,
π_{#}^{1,3}μ ∈ Opt(π_{#}^{1}γ, ν).

Proposition 4.10. Let γ ∈ 𝒫_{2}(ℝ_{1}^{d} × ℝ_{2}^{d}) be a plan and let ν ∈ 𝒫_{2}(ℝ_{3}^{d}) be a fixed measure. Then the following formula holds:

(4.13a) \frac{d}{dt}|_{t=0}W^{2}([γ](t), ν) = inf_{μ∈Opt(γ,ν)} ∥x_{2} − x_{3}∥_{μ}^{2} − ∥x_{1} − x_{2}∥_{γ}^{2} − W^{2}([γ](0), ν)
(4.13b)  = − 2 sup_{μ∈Opt(γ,ν)} ⟨x_{2} − x_{1}, x_{3} − x_{1}⟩_{μ}.

Proof. Observe that a plan μ realizes the inf in (4.13a) if and only if realizes the sup in (4.13b).

The previous lemma ensures that for any μ ∈ Opt(γ, ν) it holds

\begin{matrix}\frac{d^{+}}{dt} W^{2}([γ](t), ν)|_{t=0} ≤ & ∥x_{2} − x_{3}∥_{μ}^{2} − ∥x_{1} − x_{2}∥_{γ}^{2} − W^{2}([γ](0), ν) \\ = & − 2⟨x_{2} − x_{1}, x_{3} − x_{1}⟩_{μ}. \end{matrix}

To prove the converse inequality choose, for any t ∈ [0, 1], a 3-plan μ_{t} satisfying (4.11), observe that this family is 2-bounded in 𝒫_{2}(ℝ^{3d}) and let μ_{0} be any weak accumulation point. Remark 1.1 and proposition 1.11 give μ_{0} ∈ Opt(γ, ν), moreover from

W^{2}([γ](t), ν) = ∥(1 − t)x_{1} + tx_{2} − x_{3}∥_{μ_{t}}^{2} = ∥x_{1} − x_{3}∥_{μ_{t}}^{2} + o(1),

we get ∥x_{1} − x_{3}∥_{μ_{t}}^{2} → W^{2}([γ](0), ν) as t → 0^{+}. We conclude with:

\begin{matrix}\frac{d}{dt_{+}} |_{t=0}W^{2}([γ](t), ν) = & lim_{t→0^{+}} \frac{d}{dt_{−}} W^{2}([γ](t), ν) \\ ≥ & lim_{t→0^{+}} ∥x_{2} − x_{3}∥_{μ_{t}}^{2} − ∥x_{1} − x_{3}∥_{μ_{t}}^{2} + (2t − 1)∥x_{1} − x_{2}∥_{μ_{t}}^{2} \\ ≥ & ∥x_{2} − x_{3}∥_{μ_{0}}^{2} − W^{2}([γ](0), ν) − ∥x_{1} − x_{2}∥_{γ}^{2}. \end{matrix}

□

Before going deeper on the analysis of the properties of (𝒫_{2}(ℝ^{2d})_{μ}, W_{μ}), we open a brief parenthesis and we describe in the following subsection another possible construction of Tan_{μ}(𝒫_{2}(ℝ^{d})). As we will see, the definitions we gave are consistent with the abstract construction of tangent space for P C spaces.

3. Tangent cone for P C spaces

Recall that a length space (E, d) is positively curved in the sense of Aleksandrov if for any constant speed geodesic x_{t} : [0, 1] → E and every y ∈ E it holds

(4.14)  d^{2}(x_{t}, y) ≥ (1 − t)d^{2}(x_{0}, y) + td^{2}(x_{1}, y) − t(1 − t)d^{2}(x_{0}, x_{1}),

and that in a flat Hilbertian setting the equality always holds in the above expression.

In a Euclidean setting the angle 0 ≤ θ_{x}(y, z) ≤ π between the two segments joining x to y and z respectively, is given by the formula

cos (θ_{x}(y, z)) := \frac{⟨y − x, z − x⟩}{|y − x||z − x|} = \frac{|y − x|^{2} + |z − x|^{2} − |y − z|^{2}}{2|y − x||z − x|} ,

and it is easy to see that if y_{t} := (1 − t)x + ty, z_{t} := (1 − t)x + tz are the segment joining x to y and z respectively, then it holds

θ_{x}(y, z) = θ_{x}(y_{t}, z_{s}), ∀t, s ∈ (0, 1].

If we try to do the same in a P C space we are naturally lead to define the angle 0 ≤ θ_{x}(y, z) ≤ π from which the couple (y, z) is seen from x as:

cos (θ_{x}(y, z)) := \frac{d^{2}(x, y) + d^{2}(x, z) − d^{2}(y, z)}{2d(x, y)d(x, z)}.

3. TANGENT CONE FOR P C SPACES 49

Now let y, z : [0, 1] → E be constant speed geodesics joining x to y and z respectively, it is just a matter of calculations to see that both the functions

\begin{aligned}t & → cos (θ_{x}(y(t), z(s))), \\ s & → cos (θ_{x}(y(t), z(s))), \end{aligned}

are non increasing. It is then possible and natural to define the angle 0 ≤ θ_{x}(y, z) ≤ π between the constant speed geodesics y and z as

(4.15)  θ_{x}(y, z) := sup_{t,s∈(0,1]} θ_{x}(y(t), z(s)) = lim_{t,s→0^{+}} θ_{x}(y(t), z(s)).

In this formula the geodesics do not need to be defined in the whole interval [0, 1]: if we let [0, Ty] be the domain of y we can still define θ_{x}(y, z) as the limit of θ_{x}(y(t), z(s)) as t, s → 0^{+}, or as the supremum of θ_{x}(y(t), z(s)) among all t ∈ (0, Ty] and s ∈ (0, Tz]. Observe moreover that d(y(t), x)/t is independent on t ∈ (0, Ty] and equal to the (constant) metric derivative |ẏ| of y.

Following the analogy with the Hilbert case we define:

(4.16a)  ∥y∥_{x} := |ẏ|,
(4.16b)  ⟨y, z⟩_{x} := ∥y∥_{x}∥z∥_{x} cos (θ_{x}(y, z)),
(4.16c)  d_{x}^{2}(y, z) := ∥y∥_{x}^{2} + ∥z∥_{x}^{2} − 2⟨y, z⟩_{x}.

Let us now introduce the set Geod_{x} of all constant speed geodesics starting from x; on this set we define the equivalence relation

y ∼ z ⇔ y|_{[0,t]} = z|_{[0,t]} for some t > 0.

Since the objects defined in the formulas (4.16a), (4.16b) and (4.16c) depend only on the behavior of the geodesics near 0, it is clear that the function d_{x}(·, ·) induces a function, still denoted by d_{x}(·, ·), on Geod_{x}/ ∼: we are going to prove that this function is a distance.

Since

cos (θ_{x}(y, z)) = lim_{t,s→0^{+}} \frac{d^{2}(y(t), x) + d^{2}(z(s), x) − d^{2}(y(t), z(s))}{2d(y(t), x)d(z(t), s)}
=\frac{1}{2∥y∥_{x}∥z∥_{x}} ( lim_{t,s→0^{+}} \frac{t^{2}∥y∥_{x}^{2} + s^{2}∥z∥_{x}^{2} − d^{2}(y(t), z(s))}{ts}) ,

(4.17)

applying definitions (4.16c) we obtain

(4.18)  d_{x}(y, z) = lim_{t→0^{+}} \frac{d(y(t), z(t))}{t} = sup_{t∈[0,T]} \frac{d(y(t), z(t))}{t},

where T := min{Ty, Tz}, and therefore d_{x} is a distance as claimed.

Definition 4.11 (Tangent cone for P C spaces). The tangent cone to the space (E, d) at a point x ∈ E is the completion of Geod_{x}/ ∼ with respect to the distance d_{x}.

It is not difficult to check that the functions defined in (4.16) induce functions on Geod_{x} and that they extend continuously to Tan_{x}(E).

We now show that if we apply this abstract definition to the space 𝒫_{2}(ℝ^{d}), the result is consistent with the definition we gave in the first section.

Theorem 4.12. Let (E, d) := (𝒫_{2}(ℝ^{d}), W). Then the tangent cone just defined is canonically isometric to (Tan_{μ}(𝒫_{2}(ℝ^{d})), W_{μ}). The isometry is the unique continuous extension of the following bijective map from GEOD_{μ} to Geodμ/ ∼: a plan γ correspond to the equivalence class of the geodesic t → (π^{1} + tπ^{2})_{#}γ.

Proof. To get the claim it is sufficient to prove that for any γ_{1}, γ_{2} ∈ GEOD_{μ} it holds

(4.19)  d_{μ}(γ_{1}, γ_{2}) = W_{μ}(γ_{1}, γ_{2}).

Fix such γ_{1}, γ_{2} and think γ_{1} ∈ 𝒫_{2}(ℝ_{0}^{d} × ℝ_{1}^{d}) and γ_{2} ∈ 𝒫_{2}(ℝ_{0}^{d} × ℝ_{2}^{d}). It is clear that the value ∥x_{1}∥_{γ} is equal to ∥γ∥_{μ} defined in (4.16a), moreover we know that W_{μ}^{2} (γ_{1}, γ_{2}) = ∥x_{1} − x_{2}∥_{α}^{2} = ∥x_{1}∥_{γ_{1}}^{2} + ∥x_{2}∥_{γ_{2}}^{2} − 2⟨x_{1}, x_{2}⟩_{α}, where α ∈ OPT_{μ}(γ_{1}, γ_{2}), and therefore our thesis may be written in the following way: for any γ_{1}, γ_{2} ∈ GEODμ and any α ∈ OPT_{μ}(γ_{1}, γ_{2}) it holds

(4.20)  ⟨x_{1}, x_{2}⟩_{α} = ⟨γ_{1}, γ_{2}⟩_{μ} = ∥γ_{1}∥_{μ}∥γ_{2}∥_{μ} cos (θ_{μ}(γ_{1}, γ_{2})).

Let us examine the right hand side of this equation. Using the definitions (4.16) and equation (4.17) and taking separate limits, first w.r.t. s and then w.r.t. t, we obtain the following formula:

⟨γ_{1}, γ_{2}⟩_{μ} = − lim_{t→0^{+}} \frac{1}{2t} \frac{d}{ds}|_{s=0}(W^{2}((π^{0} + tπ^{1})_{#}γ_{1}, (π^{0} + sπ^{2})_{#}γ_{2})).

Using proposition 4.10 to compute the derivative in the right hand side and recalling (see theorem 2.8) that for small t the plan (π^{0}, π^{0} + tπ^{1})_{#}γ_{1} is the unique element of Opt(μ, (π^{0} + tπ^{1})_{#}γ_{1}), we get

(4.21)  ⟨γ_{1}, γ_{2}⟩_{μ} = lim_{t→0^{+}} \frac{1}{t}⟨x_{1} − x_{0}, x_{2} − x_{0}⟩_{α_{t}},

where α_{t} minimize ∥x_{1} −x_{2}∥_{α_{t}}^{2} among all 3-plans in ADM_{μ}((π^{0}, π^{0} +tπ^{1})_{#}γ_{1}, (π^{0}, π^{0} + π^{2})_{#}γ_{2}). Some manipulations show that α_{t} has this property if and only if the 3-plan

\overline{α}_{t} := (π^{0}, π^{1} − π^{0}, \frac{π^{2} − π^{0}}{t} )_{#} α_{t},

realizes the minimum of ∥x_{1} − x_{2}∥_{\overline{α}_{t}} among the set ADM_{μ}(γ_{1}, γ_{2}). This means that we can keep \overline{α}_{t} constant in time and equal to some \overline{α} ∈ OPT_{μ}(γ_{1}, γ_{2}) and obtain

⟨γ_{1}, γ_{2}⟩_{μ} = ⟨x_{1}, x_{2}⟩\overline{α},

and therefore our thesis is proved. □

This characterization of Tan_{μ}(𝒫_{2}(ℝ^{d})) leads us to the following definitions.

4. ON THE INCLUSION Tan_{μ}(𝒫_{2}(ℝ^{d})) ↪ Tan_{μ}(𝒫_{2}(ℝ^{d})) 51

Definition 4.13. Let γ ∈ 𝒫_{2}(ℝ^{2d})_{μ} ⊂ 𝒫_{2}(ℝ_{0}^{d} × ℝ_{1}^{d}) be a plan. The norm of γ is

∥γ∥_{μ} := ∥x_{1}∥_{γ}.

Let γ_{1}, γ_{2} ∈ 𝒫_{2}(ℝ^{2d})_{μ} be two plans. The scalar product of them is given by

2⟨γ_{1}, γ_{2}⟩ := W_{μ}^{2}(γ_{1}, γ_{2}) − ∥γ_{1}∥_{μ}^{2} − ∥γ_{2}∥_{μ}^{2} = 2⟨x_{1}, x_{2}⟩_{α}, ∀α ∈ OPT_{μ}(γ_{1}, γ_{2}).

These definitions reduce to the ones given in (4.16a), (4.16b) if the plans belong to GEOD_{μ}, here we extended them to the whole of 𝒫_{2}(ℝ^{2d})_{μ}.

4. On the inclusion Tan_{μ}(𝒫_{2}(ℝ^{d})) ↪ Tan_{μ}(𝒫_{2}(ℝ^{d}))

In this section we analyse the relationship between the two tangent spaces we defined and study the properties of the exponential map.

A particular subset of Tan_{μ}(𝒫_{2}(ℝ^{d})) is the set of all plans induced by maps: since for such plans W_{μ} essentially reduces to the more common L^{2} distance w.r.t. μ, this set is isomorphic through the embedding ℑ_{μ} to the set

Tan_{μ}(𝒫_{2}(ℝ^{d})):=\overline{\{v ∈ L^{2}(μ, ℝ^{d}) : Id + tv is optimal for some t > 0\}}^{L^{2}(μ,ℝ^{d})}.

In the first section we already introduced this set for regular measures μ, here we just dropped this assumption.

Observe that the definitions of Tan_{μ}(𝒫_{2}(ℝ^{d})) and Tan_{μ}(𝒫_{2}(ℝ^{d})) are quite different: the first one is obtained from the study of geodesics while the latter one was the result of the analysis of the continuity equation. It is therefore a non trivial fact that the following theorem holds, which shows that Tan_{μ}(𝒫_{2}(ℝ^{d})) is an enlargement of Tan_{μ}(𝒫_{2}(ℝ^{d})). Observe that there is no regularity assumption of μ.

Theorem 4.14. Let μ ∈ 𝒫_{2}(ℝ^{d}) be any measure. Then

Tan_{μ}(𝒫_{2}(ℝ^{d})) = Tan_{μ}(𝒫_{2}(ℝ^{d})).

Proof. The inclusion ⊃ is quite obvious: for any φ ∈ C_{c}^{∞}(ℝ^{d}) there exists some t > 0 such that D(Id + t∇φ) ≥ 0 in the sense of symmetric operators, this means that Id + t∇φ is the gradient of a convex function and therefore optimal.

To prove the converse implication we will show that any monotone map T ∈ L^{2}(μ, ℝ^{d}) belongs to Tan_{μ}(𝒫_{2}(ℝ^{d})). Let us first suppose that μ ≪ L^{d}, L^{d} ≪ μ, that μ has bounded density and that T is bounded, say |T (x)| ≤ C < ∞ for a.e. x ∈ ℝ^{d}. In this case for any family of mollifiers ρ_{ε} the maps T_{ε} := T ∗ ρ_{ε} are well defined, satisfy |T_{ε}(x)| ≤ C for any x ∈ ℝ^{d} and ε > 0, and converge for μ−a.e. x to T (x) as ε → 0. The fact that μ has bounded density implies that the costant functions are in L_{μ}^{2}, therefore by the dominated convergence theorem, the convergence a.e. of the equibounded functions T_{ε} implies convergence in L_{μ}^{2}. From the absolute continuity of μ and Brenier’s theorem we have the existence of a convex map φ such that T = ∇φ μ-a.e., therefore it holds T_{ε} = (∇φ) ∗ ρ_{ε} = ∇(φ ∗ ρ_{ε}) ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})). The L_{μ}^{2}−convergence of T_{ε} to T gives T ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})).

Now drop the hypothesis on the boundedness of T. Define ν := T_{#}μ and find a sequence (ν_{n}) of measures with bounded support W −converging to ν. Let T_{n} be the optimal transport map from μ to ν_{n}: by what we just proved we have T_{n} ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) for every n ∈ ℕ. Recall proposition 2.27 to obtain ∥T_{n} − T ∥_{μ} → 0 and therefore T ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})).

For the general case we can assume that ℑ_{μ}(T) is the unique optimal plan from μ to T_{#}μ: otherwise substitute T with (1 − t)Id + tT, 0 < t < 1, use proposition 2.11 to get the uniqueness of ℑ_{μ}((1 − t)Id + tT), and let t go to 1. Under this hypothesis we will use equation (3.10) and show that ⟨T, w⟩_{μ} = 0 for any w ∈ L^{2}(μ, ℝ^{d}) such that ∇ · (wμ) = 0. Let ρ_{ε} be as before a family of mollifiers and define

μ_{ε} := μ ∗ ρ_{ε},
w_{ε} := \frac{(wμ) ∗ ρ_{ε}}{μ_{ε}} .

It is clear that μ_{ε} →^{W} μ, μ_{ε} ≪ L^{d}, L^{d} ≪ μ_{ε}, sup μ_{ε}(x) ≤ sup ρ_{ε}(x), (w_{ε}, μ_{ε}) ↠ (w, μ) (use lemma 3.3 to get ∥w_{ε}∥_{μ_{ε}} ≤ ∥w∥_{μ} ) and that ∇ · (w_{ε}μ_{ε}) = 0. Moreover letting T_{ε} be the unique optimal map from μ_{ε} to T_{#}μ applying proposition 2.27 we get (T_{ε}, μ_{ε}) ↠ (T, μ). Since the previous result ensures that ⟨T_{ε}, w_{ε}⟩_{μ_{ε}} = 0 the conclusion follows by lemma 2.28 (here both sequences converge strongly). □

With this theorem we proved that the natural embedding ℑ of L_{μ}^{2} into 𝒫_{2}(ℝ^{2d})_{μ} maps the set Tan_{μ}(𝒫_{2}(ℝ^{d})) into the set Tan_{μ}(𝒫_{2}(ℝ^{d})). On the other hand we have the natural map ℬ : 𝒫_{2}(ℝ^{2d})_{μ} → L_{μ}^{2} and we may ask if the image of Tan_{μ}(𝒫_{2}(ℝ^{d})) is contained or not in Tan_{μ}(𝒫_{2}(ℝ^{d})). The answer is affirmative and given by the following theorem.

Theorem 4.15. Let γ ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})). Then ℬ(γ) ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})).

Proof. Observe that since ∥ℬ(γ_{1})−ℬ(γ_{2})∥_{μ} ≤ ∥x_{1}−x_{2}∥_{α} for any α ∈ ADM_{μ}(γ_{1}, γ_{2}), by taking the infimum w.r.t. α we get ∥ℬ(γ_{1}) − ℬ(γ_{2})∥_{μ} ≤ W_{μ}(γ_{1}, γ_{2}). Therefore in order to prove the thesis it is sufficient to consider the case of plans γ such that (π^{0}, π^{0} + tπ^{1})_{#}γ is optimal for some t > 0, or, which is the same, of plans γ which may be written as (π^{0}, 1/t(π^{1} − π^{0}))_{#}\overline{γ} for some optimal plan \overline{γ} and some t. For such plans it holds ℬ(γ) = (ℬ(\overline{γ}) − Id)/t, therefore our thesis will be proved if we show that ℬ(\overline{γ}) ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) for any optimal plan \overline{γ}. Since \overline{γ} is optimal, we know that there exists a convex function φ such that \overline{γ} is concentrated on the graph of the subdifferential of φ. Disintegrating \overline{γ} w.r.t. the first variable we obtain that supp(\overline{γ})_{x} ⊂ ∂^{−}φ(x). Given that ∂^{−}φ(x) is a closed convex set, we have that ∫ yd(\overline{γ})_{x}(y) ∈ ∂^{−}φ(x), that is: the graph of ℬ(\overline{γ}) is concentrated on the graph of the subdifferential of φ and therefore ℬ(\overline{γ}) is an optimal map. By the previous theorem we have that ℬ(\overline{γ}) ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) and therefore the thesis. □

Definition 4.16 (Multiplication of a plan with a scalar). Let γ ∈ 𝒫_{2}(ℝ^{2d})_{μ} be a plan and t ∈ ℝ. The plan t · γ ∈ 𝒫_{2}(ℝ^{2d})_{μ} is

t · γ := (π^{1}, tπ^{2})_{#}γ.

Proposition 4.17. For any t ∈ ℝ the map γ → t · γ defines an omothety on 𝒫_{2}(ℝ^{2d})_{μ} with constant |t|, i.e.

(4.22)  W_{μ}(t · γ_{1}, t · γ_{2}) = |t|W_{μ}(γ_{1}, γ_{2}).

Proof. It is sufficient to observe that for a given α ∈ ADM_{μ}(γ_{1}, γ_{2}) the plan (π^{0}, tπ^{1}, tπ^{2})_{#}α belongs to ADM_{μ}(t · γ_{1}, t · γ_{2}), and that its cost is |t| ∥x_{1} − x_{2}∥_{α}.

□

Now we characterize tangent plans through the infinitesimal behavior of the exponential map. Observe that the definition below is consistent with its analogous in L^{2}(μ, ℝ^{d}) (see 3.6).

Definition 4.18 (The exponential map and its inverse). Let γ ∈ 𝒫_{2}(ℝ^{2d})_{μ} be a plan. The exponential map exp_{μ} : 𝒫_{2}(ℝ^{2d})_{μ} → 𝒫_{2}(ℝ^{d}) is defined as

exp_{μ}(γ) := (π^{1} + π^{2})_{#}γ.

The inverse of exp_{μ} is the multivalued function from 𝒫_{2}(ℝ^{d}) to Tan_{μ}(𝒫_{2}(ℝ^{d})) defined as

γ ∈ exp_{μ}^{−1}(ν) ⇔ (π^{1}, π^{1} + π^{2})_{#}γ ∈ Opt(μ, ν).

Note that the functional inverse of exp_{μ} is different from what we defined since it may happen for a plan γ ∈ 𝒫_{2}(ℝ^{2d})_{μ} to satisfy (π^{1} + π^{2})_{#}γ = ν and (π^{1}, π^{1} + π^{2})_{#}γ ∉ Opt(μ, ν)

Note that as in the previous chapter we did not restrict the domain of exp_{μ} to tangent plans.

With this notation we have that for any γ ∈ GEOD_{μ} the map t → exp_{μ}(t · γ) is a constant speed geodesic in some right neighborhood of 0 whose metric derivative is ∥γ∥_{μ}.

Observe moreover that since α ∈ ADM_{μ}(γ_{1}, γ_{2}) implies (π^{0} + π^{1}, π^{0} + π^{2})_{#}α ∈ Adm(exp_{μ}(γ_{1}), exp_{μ}(γ_{2})) the exponential map is nonexpansive, i.e.

(4.23)  W (exp_{μ}(γ_{1}), exp_{μ}(γ_{2})) ≤ W_{μ}(γ_{1}, γ_{2}).

This is consistent with the positive curvature of (𝒫_{2}(ℝ^{d}), W).

Theorem 4.19 (Infinitesimal behavior of exp_{μ} on tangent plans). Let γ ∈ 𝒫_{2}(ℝ^{2d})_{μ} be a plan. Then γ is tangent if and only if

(4.24)  lim_{t→0^{+}} \frac{W (exp_{μ}(t · γ), μ)}{t} = ∥γ∥_{μ}.

More explicitely, if γ satisfies the previous equation, then for every choice of η_{t} ∈ Opt(μ, exp_{μ}(t · γ)) the plans γ_{t} := (π^{0}, (π^{1} − π^{0})/t)_{#}η_{t} ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) converge to γ in (𝒫_{2}(ℝ^{2d})_{μ}, W_{μ}).

Proof. We have already noticed that the γ ∈ GEOD_{μ} satisfy equation (4.24). The conclusion for general γ ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) follows by a density argument using equation (4.22) and inequality (4.23).

Let us turn to the converse implication: we will show that the plans γ_{t} defined in the statement W_{μ}−converge to γ. From the equation

W_{μ}^{2}(γ_{t}, γ) = ∥γ_{t}∥^{2} + ∥γ∥^{2} − 2⟨γ_{t}, γ⟩,

and equation (4.24) it follows that the thesis will be proved as soon as we prove that

\underline{lim}_{t→0^{+}} ⟨γ_{t}, γ⟩ ≥ ∥γ∥^{2}.

Let us evaluate the double derivative \frac{d}{ds} \frac{d}{dt}W^{2}(μ_{s}, μ_{t}) at (0, 0). By proposition 4.10 and its proof we have that

(4.25) \frac{d}{ds}|_{s=0^{+}} \frac{d}{dt}|_{t=0^{+}}W^{2}(μ_{t}, μ_{s}) = −2 lim_{s→0^{+}} \frac{1}{s}⟨γ, (π^{0}, π^{1}−π^{0})_{#}η_{s}⟩ = −2 lim_{s→0^{+}} ⟨γ, γ_{s}⟩.

On the other hand, by direct computation we have that

\frac{d}{ds}|_{s=0^{+}} \frac{d}{dt}|_{t=0^{+}}W^{2}(μ_{t}, μ_{s}) = lim_{s→0^{+}} \frac{1}{s} lim_{t→0^{+}} \frac{W^{2}(μ_{t}, μ_{s}) − W^{2}(μ_{0}, μ_{s})}{t}
= lim_{s→0^{+}} lim_{t→0^{+}} \frac{W (μ_{t}, μ_{s}) − W (μ_{0}, μ_{s})}{t} \frac{W (μ_{t}, μ_{s}) + W (μ_{0}, μ_{s})}{s}
=2 lim_{s→0^{+}} lim_{t→0^{+}} \frac{W (μ_{t}, μ_{s}) − W (μ_{0}, μ_{s})}{t} lim_{s→0^{+}} \frac{W (μ_{0}, μ_{s})}{s}.

Applying the trivial inequality W (μ_{t}, μ_{s}) ≤ (s − t)∥γ∥, valid for all 0 ≤ t ≤ s, and recalling equation (4.24) it follows that

(4.26)  \frac{d}{ds}|_{s=0^{+}} \frac{d}{dt}|_{t=0^{+}}W^{2}(μ_{t}, μ_{s}) ≤ −2∥γ∥^{2}.

Putting together equations (4.25) and (4.26) we get lim_{s→0^{+}} −2⟨γ, γ_{s}⟩ ≤ −2∥γ∥^{2} as de-

sired. □

5. Operations on 𝒫_{2}(ℝ^{2d})_{μ}

For general measures the geometric tangent space is different from the regular one and it is not a vector space. However, it is still possible to define the product of a plan with a real number (we already defined it in 4.16), a sum between plans and a scalar product (defined in 4.13) giving to this space a structure similar to the one of an Hilbert space. However the sum is a multivalued operator.

Definition 4.20. Let γ_{1}, γ_{2} ∈ 𝒫_{2}(ℝ^{2d})_{μ} be two plans and α ∈ ADM_{μ}(γ_{1}, γ_{2}). The sum of γ_{1} and γ_{2} through α is

γ_{1} +_{α} γ_{2} := (π^{0}, π^{1} + π^{2})_{#}α.

The set of all the possible values of γ_{1} +_{α} γ_{2} as α varies in ADM_{μ}(γ_{1}, γ_{2}) is denoted by γ_{1} ⊕ γ_{2}.

We recall that for a given couple γ_{1}, γ_{2} ∈ 𝒫_{2}(ℝ^{2d})_{μ} of plans, their scalar product is defined as

(4.27) 2⟨γ_{1}, γ_{2}⟩ := ∥γ_{1}∥^{2} + ∥γ_{2}∥^{2} − W_{μ}^{2}(γ_{1}, γ_{2}) = 2⟨x_{1}, x_{2}⟩_{α},  α ∈ OPT_{μ}(γ_{1}, γ_{2}).

5. OPERATIONS ON 𝒫_{2}(ℝ^{2d})_{μ} 55

An equivalent definition is given by the following formula:

⟨γ_{1}, γ_{2}⟩ := sup_{α′∈ADM_{μ}(γ_{1},γ_{2})} ⟨x_{1}, x_{2}⟩_{α_{′}}.

In this definition we took the supremum (maximum) of ⟨x_{1}, x_{2}⟩_{α_{′}}, letting the inner product be a single valued operator. A different choice would be to deal with the multivalued operator given by the whole set of ⟨x_{1}, x_{2}⟩_{α_{′}} as α varies in ADM_{μ}(γ_{1}, γ_{2}): in this case the operator has better linearity properties (for instance the thesis of proposition 4.27 would be true even for λ < 0). Choosing one definition or the other is mainly a matter of taste. We preferred to keep our definition because it is consistent with the scalar product on the tangent cone for P C spaces (see equation (4.16b), (4.16c)), but notice that here we defined the product for any couple of plans in 𝒫_{2}(ℝ^{2d})_{μ} and not only for those in Tan_{μ}(𝒫_{2}(ℝ^{d})).

Observe that a simple consequence of equation (4.27) is the continuity of the inner product. A better result is given by the following proposition.

Proposition 4.21. Let γ_{1}, γ_{2}, η ∈ 𝒫_{2}(ℝ^{2d})_{μ} be given plans. Then
|⟨γ_{1}, η⟩ − ⟨γ_{2}, η⟩| ≤ W_{μ}(γ_{1}, γ_{2})∥η∥

Proof. Let α_{1} ∈ OPT_{μ}(γ_{1}, η), and α_{2} ∈ OPT_{μ}(γ_{1}, γ_{2}), be two optimal 3-plan, and consider a 4-plan β satisfying

\begin{aligned}π_{#}^{0,1,2}β & = α_{1}, \\ π_{#}^{0,1,3}β & = α_{2}. \end{aligned}

Then we have π_{#}^{0,2,3}β ∈ ADM_{μ}(η, γ_{2}) and therefore

⟨γ_{1}, η⟩ = ⟨x_{1}, x_{2}⟩_{β} = ⟨x_{1} − x_{3}, x_{2}⟩_{β} + ⟨x_{3}, x_{2}⟩_{β} ≤ W_{μ}(γ_{1}, γ_{2})∥η∥ + ⟨γ_{2}, η⟩.

By exchanging the roles of γ_{1} and γ_{2} the thesis is proven. □

In the next propositions we show that sum and product with scalar give a structure similar to that of a vectorial space.

Proposition 4.22. Let λ_{1}, λ_{2} ∈ ℝ, μ ∈ 𝒫_{2}(ℝ^{d}) be a measure and γ_{i} ∈ 𝒫_{2}(ℝ^{2d}), i = 1, 2, 3, be plans such that π_{#}^{1}γ_{i} = μ, i = 1, 2, 3. Then we have:

(4.28a)  γ_{1} ⊕ γ_{2} = γ_{2} ⊕ γ_{1},
(4.28b)  γ_{1} ⊕ (γ_{2} ⊕ γ_{3}) = (γ_{1} ⊕ γ_{2}) ⊕ γ_{3},
(4.28c)  0_{μ} ⊕ γ_{1} = {γ_{1}},
(4.28d)  0 · γ_{1} = 0_{μ},
(4.28e)  1 · γ_{1} = γ_{1},
(4.28f)  λ_{1} · (γ_{1} ⊕ γ_{2}) = (λ_{1} · γ_{1}) ⊕ (λ_{1} · γ_{2}),
(4.28g)  λ_{1} · (λ_{2} · γ_{1}) = (λ_{1}λ_{2}) · γ_{1},
(4.28h)  (λ_{1} + λ_{2}) · γ_{1} ⊂ (λ_{1} · γ_{1}) ⊕ (λ_{2} · γ_{1}),

where 0_{μ} := (Id, 0)_{#}μ ∈ 𝒫_{2}(ℝ^{2d}).

Proof. (4.28a), (4.28d), (4.28e), (4.28g) are obvious; since ADM_{μ}(γ_{1}, 0_{μ}) contains only the plan (π^{1}, π^{2}, 0)_{#}γ_{1} we obtain (4.28c). If λ_{1} = 0 then (4.28f) simply says 0 = 0⊕0_{μ}, for the other case note that α ∈ ADM_{μ}(γ_{1}, γ_{2}) if and only if (π^{1}, λ_{1}π^{2}, λ_{1}π^{3})_{#}α ∈ ADM_{μ}(λ_{1} · γ_{1}, λ_{1} · γ_{2}). (4.28h) follows observing that the plan (π^{1}, λ_{1}π^{2}, λ_{2}π^{2})_{#}γ_{1} belongs to ADM_{μ}(λ_{1} · γ_{1}, λ_{2} · γ_{1}). It remains to prove (4.28b), in order to make the proof more clear we think γ_{i} ∈ 𝒫_{2}(ℝ_{0}^{d} × ℝ_{i}^{d}). Choose α_{1} ∈ ADM_{μ}(γ_{2}, γ_{3}), α_{2} ∈ ADM_{μ}(γ_{1}, γ_{2} +_{α_{1}} γ_{3}) and define α′_{1} := (π^{0}, π^{1} + π^{2}, π^{2})_{#}α_{1}. Dudley’s lemma ensures the existence of a 4-plan α such that π_{#}^{0,1,2}α = α_{2} and π_{#}^{0,2,3}α = α′_{1}; it follows that the plan \overline{α} := (π^{0}, π^{1}, π^{2} − π^{3}, π^{3})_{#}α satisfies π_{#}^{0,i}\overline{α} = γ_{i}, i = 1, 2, 3, and defining β_{1} := (π^{0}, π^{1}, π^{2})_{#}\overline{α}, β_{2} := (π^{0}, π^{1} + π^{2}, π^{3})_{#}\overline{α} we obtain

γ_{1} +_{α_{2}} (γ_{2} +_{α_{1}} γ_{3}) = (γ_{1} +_{β_{1}} γ_{2}) +_{β_{2}} γ_{3}.

This proves one inclusion, the other is analogous. □

Remark 4.23. In general the inclusion in (4.28h) is strict: indeed if the plan γ is not induced by a map the set ADM_{μ}(λ_{1} · γ, λ_{2} · γ) does not contain only the plan (π^{1}, λ_{1}π^{2}, λ_{2}π^{2})_{#}γ. The following is an explicit example: let μ := δ_{0} ∈ 𝒫_{2}(ℝ) be the base measure and γ := 1/2(δ_{(0,0)} + δ_{(0,1)}) ∈ 𝒫_{2}(ℝ^{2})_{μ}. Then 2 · γ = 1/2(δ_{(0,0)} + δ_{(0,2)}) and

γ ⊕ γ = {1/2(λδ(0, 0) + (2 − 2λ)δ_{(0,1)} + λδ_{(0,2)}) : λ ∈ [0, 1]}.

Proposition 4.24 (continuity of the operations). Let μ ∈ 𝒫_{2}(ℝ^{d}) be a measure and let γ ∈ 𝒫_{2}(ℝ^{2d})_{μ} be a plan. Then the map λ → λ · γ is Lipschitz continuous from ℝ to 𝒫_{2}(ℝ^{2d})_{μ} and the map (γ_{1}, γ_{2}) → γ_{1} ⊕ γ_{2} is continuous from (𝒫_{2}(ℝ^{2d})_{μ})^{2} to the set of compact subsets of 𝒫_{2}(ℝ^{2d})_{μ} endowed with the Hausdorff distance.

Proof. The first assertion follows by noticing that (π^{0}, λ_{1}π^{1}, λ_{2}π^{2})_{#}γ is an admissible 3-plan for the couple (λ_{1} · γ, λ_{2} · γ) and its cost is |λ_{1} − λ_{2}∥|γ∥. In order to prove the second one let us first observe that the compactness of γ_{1} ⊕ γ_{2} follows from the compactness of ADM_{μ}(γ_{1}, γ_{2}) and the continuity of the map α → (π^{0}, π^{1} + π^{2})_{#}α. Now fix γ_{1}, γ_{2}, γ′_{1}, γ′_{2} ∈ 𝒫_{2}(ℝ^{2d})_{μ}, we will show that for every γ ∈ γ_{1} ⊕ γ_{2} there exists γ′ ∈ γ′_{1} ⊕ γ′_{2} such that W_{μ} (γ, γ′) ≤ W_{μ} (γ_{1}, γ′_{1}) + W_{μ} (γ_{2}, γ′_{2}): choose α_{i} ∈ OPT_{μ}(γ_{i}, γ′_{i}), i = 1, 2, γ ∈ γ_{1} ⊕ γ_{2} and α ∈ ADM_{μ}(γ_{1}, γ_{2}) such that γ = γ_{1} +_{α} γ_{2} and find, thanks to Dudley’s lemma, a 5-plan β such that

π_{#}^{0,1,2}β = α_{1}
π_{#}^{0,3,4}β = α_{2}
π_{#}^{0,1,3}β = α.

Defining α′ := π_{#}^{0,2,4}β, γ′ := γ′_{1} +_{α}′ γ′_{2} and \overline{β} := (π^{0}, π^{1} + π^{3}, π^{2} + π^{4})_{#}β we obtain:

W_{μ} (γ, γ′) ≤ ∥x_{1} − x_{2}∥\overline{β} = ∥x_{1} + x_{3} − x_{2} − x_{4}∥_{β} ≤
∥x_{1} − x_{2}∥_{β} + ∥x_{3} − x_{4}∥_{β} = W_{μ} (γ_{1}, γ′_{1}) + W_{μ} (γ_{2}, γ′_{2}) .

□

These operations and their properties are very similar to those which define the vector space structure, now we want to prove that for any measure μ ∈ 𝒫_{2}(ℝ^{d}) the geometric tangent space Tan_{μ}(𝒫_{2}(ℝ^{d})) is a “subspace” of 𝒫_{2}(ℝ^{2d})_{μ}, i.e. we will prove that Tan_{μ}(𝒫_{2}(ℝ^{d})) is stable under these operations.

Proposition 4.25. For every γ_{1}, γ_{2} ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) we have γ_{1} ⊕ γ_{2} ⊂ Tan_{μ}(𝒫_{2}(ℝ^{d})).

Proof. By proposition 4.24 it is enough to prove the theorem under the additional hypothesis γ_{1}, γ_{2} ∈ GEOD_{μ}, so we assume that there exists t > 0 such that (π^{1}, π^{1} + tπ^{2})_{#}γ_{i}, i = 1, 2, is optimal. This means that for i = 1, 2, n ∈ ℕ, σ permutation of {1, …, n} and (x_{1}, y_{1}), …, (x_{n}, y_{n}) ∈ supp γ_{i} it holds

\begin{matrix}∑_{j=1}^{n} |x_{j}|^{2} + t ∑_{j=1}^{n} ⟨x_{j}, y_{j}⟩ = & ∑_{j=1}^{n} ⟨x_{j}, x_{j} + ty_{j}⟩ \\ ≥ & ∑_{j=1}^{n} ⟨x_{j}, x_{σ(j)} + ty_{σ(j)}⟩ = ∑_{j=1}^{n} ⟨x_{j}, x_{σ(j)}⟩ + t ∑_{j=1}^{n} ⟨x_{j}, y_{σ(j)}⟩. \end{matrix} (4.29)

Choose α ∈ ADM_{μ}(γ_{1}, γ_{2}) and define γ := γ_{1} +α γ_{2}: we want to prove that (π^{1}, π^{1} + t/2π^{2})_{#}γ is optimal, so we fix n ∈ ℕ, σ ∈ S_{n} and (x_{j}, y_{j}) ∈ supp γ, j = 1, …, n and we will show that it holds

∑_{j=1}^{n} |x_{j}|^{2} + t/2 ∑_{j=1}^{n} ⟨x_{j}, y_{j}⟩ ≥ ∑_{j=1}^{n} ⟨x_{j}, x_{σ(j)}⟩ + t/2 ∑_{j=1}^{n} ⟨x_{j}, y_{σ(j)}⟩.

We know that for every ε > 0 and every (x, y) ∈ supp γ there exists (x′, y^{1}, y^{2}) ∈ supp α such that |(x, y) − (x′, y^{1} + y^{2})| < ε, in particular (x′, y^{i}) ∈ supp γ_{i}, i = 1, 2. Choosing in this way (x′_{j}, y_{j}^{1}, y_{j}^{2}), j = 1, …, n, from (x_{j}, y_{j}), j = 1, …, n, and adding up the two inequalities (4.29) relatives to γ_{1} and γ_{2} obtained with these (x′_{j}, y_{j}^{1}) and (x′_{j}, y_{j}^{2}), j = 1, …, n we get

\begin{matrix}2 ∑_{j=1}^{n} |x_{j}|^{2} + t ∑_{j=1}^{n} ⟨x_{j}, y_{j}⟩ ≥ & 2 ∑_{j=1}^{n} |x′_{j}|^{2} + t ∑_{j=1}^{n} ⟨x′_{j}, y_{j}^{1} + y_{j}^{2}⟩ − 8Mnε \\ ≥ & 2 ∑_{j=1}^{n} ⟨x′_{j}, x′_{σ(j)}⟩ + t ∑_{j=1}^{n} ⟨x_{j}, y_{σ(j)}^{1} + y_{σ(j)}^{2}⟩ − 8Mnε \\ ≥ & 2 ∑_{j=1}^{n} ⟨x_{j}, x_{σ(j)}⟩ + t ∑_{j=1}^{n} ⟨x_{j}, y_{σ(j)}⟩ − 16Mnε, \end{matrix}

where M := max_{j} |(x_{j}, y_{j})|. Letting ε go to 0 we get the thesis. □

Remark 4.26. A similar computation shows that if γ_{1}, γ_{2} are optimal plans with the same first marginal, then each element of γ_{1} ⊕ γ_{2} is an optimal plan, too. Since in addition if γ is optimal then even λ · γ is optimal we get that any linear combination (w.r.t. the operations just defined) with positive coefficients of optimal plans is optimal.

Proposition 4.27. Let γ_{1}, γ_{2}, γ_{3} ∈ 𝒫_{2}(ℝ^{2d})_{μ} and λ ≥ 0. Then it holds

⟨λ · γ_{1}, γ_{2}⟩ = ⟨γ_{1}, λ · γ_{2}⟩ = λ⟨γ_{1}, γ_{2}⟩, (4.30) ⟨γ_{1}, γ_{3}⟩ + ⟨γ_{2}, γ_{3}⟩ = max ⟨γ_{1} ⊕ γ_{2}, γ_{3}⟩. (4.31)

Proof. Equation (4.30) is clear if λ = 0, otherwise it is sufficient to observe that α ∈ OPT_{μ}(γ_{1}, γ_{2}) iff (π^{0}, λπ^{1}, π^{2})_{#}α ∈ OPT_{μ}(λ · γ_{1}, γ_{2}) and iff (π^{0}, π^{1}, λπ^{2})_{#}α ∈ OPT_{μ}(γ_{1}, λ · γ_{2}).

To prove (4.31) let us first prove that the sum on the left is an element of the set on the right: observe that for every choice of α_{i,3} ∈ OPT_{μ}(γ_{i}, γ_{3}), i = 1, 2, there exists a 4-plan β satisfying

π_{#}^{0,1,3}β = α_{1,3},
π_{#}^{0,2,3}β = α_{2,3},

Define α_{1,2} := π_{#}^{0,1,2}β and check that a consequence of the previous remark is

(π^{0}, π^{1} + π^{2}, π^{3})_{#}β ∈ OPT_{μ}(γ_{1} +_{α_{1,2}} γ_{2}, γ_{3}),

from which it follows the claim. To prove that it holds (4.31), choose any element \overline{γ} ∈ γ_{1} ⊕ γ_{2}, say \overline{γ} = γ_{1} +α γ_{2}, and find a 4-plan β satisfying

π_{#}^{0,1,2}β = α,
(π^{0}, π^{1} + π^{2}, π^{3})_{#}β ∈ OPT_{μ}(\overline{γ}, γ_{3}),

then conclude with

⟨\overline{γ}, η⟩ = ⟨x_{1} + x_{2}, x_{3}⟩_{β} = ⟨x_{1}, x_{3}⟩_{β} + ⟨x_{2}, x_{3}⟩_{β} ≤ ⟨γ_{1}, η⟩ + ⟨γ_{2}, η⟩.
□

Remark 4.28. In general it is not true that ⟨λ · γ, η⟩ = λ⟨γ, η⟩, for a negative λ. Think for instance to the case λ = −1: we have that

α ∈ ADM_{μ}(γ, η) ⇔ \overline{α} := (π^{0}, −π^{1}, π^{2})_{#}α ∈ ADM_{μ}(−1 · γ, η),

and therefore it it holds

−1⟨ − 1 · γ, η⟩ = − (  max_{\overline{α}∈ADM_{μ}(−1·γ,η)} ⟨x_{1}, x_{2}⟩\overline{α})
= min_{\overline{α}∈ADM_{μ}(−1·γ,η)} ⟨ − x_{1}, x_{2}⟩\overline{α} = min_{α∈ADM_{μ}(γ,η)} ⟨x_{1}, x_{2}⟩_{α}.

Since in general min_{α}⟨x_{1}, x_{2}⟩_{α} < max_{α}⟨x_{1}, x_{2}⟩_{α}, when α varies in ADM_{μ}(γ, η) we get the claim.

It is obvious from the definition that if γ ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) then λ · γ ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) for any λ > 0, so the question is whether the same property holds for λ < 0. The answer is given by the following proposition.

Proposition 4.29. Let γ ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) be a plan. Then −1 · γ ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})).

Proof. By density and positive homogeneity we can assume that \overline{γ} := (π^{1}, π^{1} + π^{2})_{#}γ is optimal. Since

−1 · γ = −1 · \overline{γ} ⊕ (Id, Id)_{#}μ,

and since (Id, Id)_{#}μ clearly belongs to Tan_{μ}(𝒫_{2}(ℝ^{d})), by proposition 4.25 our thesis become: the opposite of an optimal plan belongs to the geometric tangent space.

So let γ be an optimal plan, we know that there exists a convex function ψ such that supp γ is contained in the graph of the subdifferential of ψ. Clearly −1 · γ has the support contained in the graph of the superdifferential of the concave function −ψ.

Since we want to prove that −1 · γ ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) we need to find a sequence of plans in GEOD_{μ} which tends to −1 · γ.

Observe that a plan η ∈ 𝒫_{2}(ℝ^{2d})_{μ} belongs to GEOD_{μ} if and only if for some t > 0 the support of (π^{1} +tπ^{2})_{#}η is cyclically monotone. By Rockafellar theorem this is true if and only if there exists a convex function φ such that (π^{1} + tπ^{2})_{#}η is concentrated on the graph of ∂^{−}φ, and therefore if and only if η is concentrated on the graph of ∂^{−}((φ − Id)/t). Recalling that a function ϕ is said semi-convex if it exists some t > 0 such that Id + tϕ is convex, we just proved that η ∈ GEOD_{μ} if and only if it is concentrated on the graph of the subdifferential of a semi-convex function.

Therefore part of the problem is to obtain the superdifferential of the concave function −ψ as limit (in some sense) of subdifferential of semi-convex functions. The key idea here is the following statement: for any convex set C ⊂ ℝ^{d}, any interior point (in the sense of convex sets) p ∈ C it holds −C ⊂ ∪_{n=0}^{∞}(nC − (n + 1)p). This is true because letting C̃ := C − p the assertion becomes −C̃ ⊂ ∪_{n=0}^{∞}nC̃ which is obviously because the fact that 0 is an interior point of C̃ implies that ∪_{n=0}^{∞}nC̃ is the whole linear span of C̃.

From this inclusion we can hope for an inclusion like ∂^{+}(−ψ)(x_{0}) ⊂ ∂^{−}(nψ − (n + 1)φ)(x_{0}), where φ is a regular function such that ∇φ(x_{0}) is an interior point of ∂^{−}ψ(x_{0}) for μ−a.e. x_{0}. However this cannot be done because such a function φ may not exist, so we have to proceed in two steps. In the first one we approximate −1 · γ with measures η_{n} whose disintegration (η_{n})_{x_{0}} are such that

supp(η_{n})_{x_{0}} ⊂ n∂^{−}ψ − (n + 1)ℬ(γ)(x_{0}),

for μ−a.e. x_{0}. In the second step we approximate ℬ(γ) with the gradient of regular functions φ_{m} and show that for any n there exists a sequence (η_{n,m}) of measures which converges to η_{n} such that supp η_{n,m} ⊂ ∂^{−}(nψ − (n + 1)φ_{m}).

Now we turn to the details. step 1 Let x_{0} → γ_{x_{0}} be the disintegration of γ with respect to μ and, for every x_{0}, let

A_{x_{0}} ⊂ ℝ^{d} be the closed convex hull of supp γ_{x_{0}}. It is clear that A_{x_{0}} ⊂ ∂^{−}ψ(x_{0}) for μ−a.e. x_{0} and that ℬ(γ)(x_{0}) is either the only point of A_{x_{0}} (if γ_{x_{0}} is not a Dirac mass) or an inteorior point of A_{x_{0}} in the sense of convex sets. Our previous statement about convex sets (with C = A_{x_{0}} and p = ℬ(γ)(x_{0})) gives that for almost every x_{0} it holds

−A_{x_{0}} ⊂ ⋃_{n=0}^{∞} (nA_{x_{0}} − (n + 1)ℬ(γ)(x_{0})),

the inclusion being obvious if A_{x_{0}} = {ℬ(γ)(x_{0})}. Let r_{n}(x_{0}) be the function defined as

r_{n}(x_{0}) := sup{r : B_{r}(−ℬ(γ)(x_{0})) ⊂ nA_{x_{0}} − (n + 1)ℬ(γ)(x_{0})},

if γ_{x_{0}} is not a Dirac mass, and r_{n}(x_{0}) := +∞ elsewhere. It is clear that

(4.32)  r_{n}(x_{0}) ⟶^{n} +∞ for μ − a.e. x_{0}.

Let P^{r,x_{0}} : ℝ^{d} → ℝ^{d} be the projection onto the ball \overline{B}_{r}(−ℬ(γ)(x_{0})) and define

η_{n,x_{0}} := (P^{r_{n}(x_{0}),x_{0}})_{#}γ_{x_{0}}.

and η_{n} := μ ⊗ η_{n,x_{0}}. Obviously it holds

supp η_{n,x_{0}} ⊂ nA_{x_{0}} − (n + 1)ℬ(γ)(x_{0}),

for any n and μ−a.e. x_{0}. Equation (4.32) implies that for μ−a.e. x_{0} it holds W (η_{n,x_{0}}, γ_{x_{0}}) → 0 as n → ∞, moreover we have

\begin{matrix}W^{2}(η_{n,x_{0}}, γ_{x_{0}}) ≤ & ∫ |P^{r_{n}(x_{0}),x_{0}}(x_{1}) − x_{1}|^{2}dγ_{x_{0}}(x_{1}) \\ ≤ & ∫ |x_{1} − ℬ(γ)(x_{0})|^{2}dγ_{x_{0}}(x_{1}) ≤ ∫ |x_{1}|^{2}dγ_{x_{0}}(x_{1}). \end{matrix}

Since ∫ |x_{1}|^{2}dγ_{x_{0}}(x_{1})dμ(x_{0}) = ∫ |x_{1}|^{2}dγ(x_{0}, x_{1}) < +∞ the dominated convergence theorem gives

W_{μ}^{2} (η_{n}, γ) = ∫ W^{2}(η_{n,x_{0}}, γ_{x_{0}})dμ(x_{0}) → 0.

step 2 We know (see theorem 4.15) that ℬ(γ) ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})), so there exists a sequence (φ_{m}) of functions in C_{c}^{∞} such that ∇φ_{m} → ℬ(γ) in L_{μ}^{2}. Consider for any n, m, x_{0} the translation defined as

τ^{n,m,x_{0}}(x_{1}) := x_{1} + (n + 1)(ℬ(γ)(x_{0}) − ∇φ_{m}(x_{0})),

and define η_{n,m,x_{0}} := (τ^{n,m,x_{0}})_{#}η_{n,x_{0}}. Since

τ^{n,m,x_{0}}(nA_{x_{0}} − (n + 1)ℬ(γ)(x_{0})) = nA_{x_{0}} − (n + 1)∇φ_{m}(x_{0}),

we have

supp η_{n,m,x_{0}} ⊂ nA_{x_{0}} − (n + 1)∇φ_{m}(x_{0}) ⊂ ∂^{−}(nψ − (n + 1)φ_{m})(x_{0}),

and so the measures η_{n,m} := μ ⊗ η_{n,m,x_{0}} belong to GEOD_{μ}. Fix n and observe that

\begin{matrix}W_{μ}^{2} (η_{n,m}, η_{n}) = & ∫ W^{2}(η_{n,m,x_{0}}, η_{n,x_{0}})dμ(x_{0}) \\ ≤ & (n + 1)∥ℬ(γ) − ∇φ_{m}∥_{μ}^{2} → 0, as m → ∞. \end{matrix}

From this fact and the convergence of η_{n} to γ in Tan_{μ}(𝒫_{2}(ℝ^{d}))(𝒫_{2}(ℝ^{d})) it follows the existence of a sequence (m_{n}) such that W_{μ} (η_{n,m_{n}}, γ) → 0 as n → ∞, and the proof is complete.

□

6. THE PROJECTION ONTO Tan_{μ}(𝒫_{2}(ℝ^{d})) 61

6. The projection onto Tan_{μ}(𝒫_{2}(ℝ^{d}))

In this section we show that there is a well defined projection operator from 𝒫_{2}(ℝ^{2d})_{μ} to Tan_{μ}(𝒫_{2}(ℝ^{d})), and we describe its basic properties.

The first proposition below asserts that there is a well defined projection operator from 𝒫_{2}(ℝ^{2d})_{μ} to Tan_{μ}(𝒫_{2}(ℝ^{d})). Since its proof is very similar to the one which says that there is a unique projection onto a closed and convex subset of an Hilbert space, in order to better follow the argument we will report first the proof of this well known fact.

Hilbert case 1. Let K be a closed convex subset of an Hilbert space H. Then for each vector v ∈ H there exists a unique P (v) ∈ K which minimizes the distance from v among all w ∈ K.

Proof. Let (w_{n}) be a minimizing sequence and let d := inf_{w∈K} |v − w|. Since (w_{n} + w_{m})/2 ∈ K by the parallelogram rule we have

\begin{matrix}|\frac{w_{n} − w_{m}}{2} |^{2} = & \frac{1}{2}|v − w_{n}|^{2} + \frac{1}{2}|v − w_{m}|^{2} − |v − \frac{w_{n} + w_{m}}{2} |^{2} \\ ≤ & \frac{1}{2}|v − w_{n}|^{2} + \frac{1}{2}|v − w_{m}|^{2} − d^{2}, \end{matrix} ; (4.33)

from which it follows that (w_{n}) is a Cauchy sequence. From the minimizing property of (w_{n}), we get that its limit produces the (only) minimizer vector P (v). □

Proposition 4.30 (projection onto the tangent space). Let γ ∈ 𝒫_{2}(ℝ^{2d})_{μ} be a plan. Then there exists a unique plan P(γ) ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) (called projection of γ onto the tangent space) which minimizes the distance W_{μ} (γ, η) among all η ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})).

Proof. Let (η_{n}) be a minimizing sequence and let d := inf W_{μ} (γ, η) among all η ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})). For each n, m ∈ ℕ let α_{n,m} ∈ 𝒫_{2}(ℝ^{4d}) be a 4-plan such that

π_{#}^{0,1,2}α_{n,m} ∈ OPT_{μ}(γ, η_{n}),
π_{#}^{0,1,3}α_{n,m} ∈ OPT_{μ}(γ, η_{m}),

and define β_{n,m} := π_{#}^{0,2,3}α_{n,m} and η_{n,m} := 1/2 · (η_{n} +_{β_{n,m}} η_{m}). We know that η_{n,m} ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) so W_{μ} (γ, η_{n,m}) ≥ d and therefore

\begin{matrix}\frac{1}{4}W_{μ}^{2} (η_{n}, η_{m}) ≤ & \frac{1}{4}∥x_{2} − x_{3}∥_{α_{n,m}}^{2} \\ ≤ & \frac{1}{2}∥x_{1} − x_{2}∥_{α_{n,m}}^{2} + \frac{1}{2}∥x_{1} − x_{3}∥_{α_{n,m}}^{2} − ||x_{1} − \frac{x_{2} + x_{3}}{2} ||_{α_{n,m}}^{2} \\ ≤ & \frac{1}{2}W_{μ}^{2} (γ, η_{n}) + \frac{1}{2}W_{μ}^{2} (γ, η_{m}) − W_{μ}^{2} (γ, η_{n,m}) \\ ≤ & \frac{1}{2}W_{μ}^{2} (γ, η_{n}) + \frac{1}{2}W_{μ}^{2} (γ, η_{m}) − d^{2}, \end{matrix}

and letting n, m → ∞ we obtain that (η_{n}) is a Cauchy sequence. □

Now we are going to prove that there exists a unique optimal 3-plan between γ and P(γ) and that this plan is induced by map (see equation (4.36) for the definition). Before doing this we need the following lemma. In the sequel we denote by \overline{Conv} A the closed convex hull of A ⊂ ℝ^{d}.

Lemma 4.31. Let γ, η ∈ 𝒫_{2}(ℝ^{2d})_{μ} be two plans and suppose that γ ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) and that for μ-a.e. x_{0} it holds

(4.34)  supp(η_{x_{0}}) ⊂ \overline{Conv} ( supp(γ_{x_{0}})).

Then η ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})).

Proof. Since γ ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) we know that there exists a sequence (γ^{n}) ∈ GEOD_{μ} converging to γ w.r.t. W_{μ}. Up to extracting a subsequence, not relabeled, we may assume that for μ−a.e. x_{0} it holds W (γ_{x_{0}}^{n}γ_{x_{0}}) → 0 as n → ∞ and that the sequence of functions x_{0} → ∥γ_{x_{0}}^{n}∥ is uniformly dominated by an L^{2} function g.

Now observe that since γ^{n} ∈ GEOD_{μ}, there exists a sequence of semi-convex functions (φ^{n}) such that for any n and μ-a.e. x_{0} it holds supp(γ_{x_{0}}^{n}) ⊂ ∂^{−}φ^{n}(x_{0}) (see also the proof of proposition 4.29). Given that ∂^{−}φ^{n}(x_{0}) is closed and convex for any n ∈ ℕ, x_{0} ∈ ℝ^{d}, it holds

\overline{Conv} ( supp (γ_{x_{0}}^{n})) ⊂ ∂^{−}φ^{n}(x_{0}), for μ-a.e. x_{0}.

Consider the projections P_{x_{0}}^{n} : ℝ^{d} → ∂^{−}φ^{n}(x_{0}) and observe that the map (x_{0}, x_{1}) → P_{x_{0}}^{n}(x_{1}) is measurable for every n ∈ ℕ. Define, for every n ∈ ℕ, the plan \overline{γ}^{n} ∈ 𝒫_{2}(ℝ^{2d})_{μ} by defining its disintegration w.r.t. the first marginal by (\overline{γ}^{n})_{x_{0}} := (P_{x_{0}}^{n})_{#}γ_{x_{0}}. Estimating W ((\overline{γ}^{n})_{x_{0}}, γ_{x_{0}}) with ∥P_{x_{0}}^{n} − Id∥_{γ_{x_{0}}} and using the minimizing properties of the projection and the fact that supp(γ_{x_{0}}^{n}) ⊂ ∂^{−}φ^{n}(x_{0}), we obtain W ((\overline{γ}^{n})_{x_{0}}, γ_{x_{0}}) ≤ W (γ_{x_{0}}^{n}, γ_{x_{0}}). Therefore (\overline{γ}^{n}) W_{μ}−converges to γ. As a consequence, we get

(4.35)  |(x_{0}P_{x_{0}}^{n}(x_{1})) − (x_{0}, x_{1})| → 0, as n → ∞ for γ-a.e. (x_{0}, x_{1}).

Analogously, define the plans η^{n} ∈ 𝒫_{2}(ℝ^{2d})_{μ} by defining their disintegrations w.r.t. their first marginal by (η^{n})_{x_{0}} := (P_{x_{0}}^{n})_{#}η_{x_{0}}. Given that clearly supp((η^{n})_{x_{0}}) ⊂ ∂^{−}φ^{n}(x_{0}), it holds η^{n} ∈ GEOD_{μ}. Therefore our thesis will be proved if we show that W_{μ}(η^{n}, η) → 0 as n → ∞.

Inclusion (4.34), the convexity of ∂^{−}φ^{n}(x_{0}), (4.35) and the convexity of (x, y) → |x − y| give

|P_{x_{0}}^{n}(x_{1}) − (x_{0}, x_{1})| → 0, as n → ∞ for η-a.e. (x_{0}, x_{1}).

To conclude it is sufficient to show that the sequence of functions x_{0} → W (η_{x_{0}}, η_{x_{0}}^{n}) is dominated by a function in L_{μ}^{2}. To this aim define the functions f^{n} : supp(μ) → ℝ^{d} letting f^{n}(x_{0}) be the element of minimal norm of ∂^{−}φ^{n}(x_{0}). Since γ^{n} ∈ 𝒫_{2}(ℝ^{2d})_{μ} and supp(γ_{x_{0}}^{n}) ⊂ ∂^{−}φ^{n}(x_{0}), we have that (f^{n}) is a sequence in L_{μ}^{2} dominated by the function g. Conclude estimating from above W (η_{x_{0}}, η_{x_{0}}^{n}) with the cost of the transport map P_{x_{0}}^{n}:

(∫ |x_{1} − P_{x_{0}}^{n}(x_{1})|^{2}dη_{x_{0}}(x_{1}))^{1/2} ≤ (∫ |x_{1} − f_{n}(x_{0})|^{2}dη_{x_{0}}(x_{1}))^{1/2}

≤ (∫ |x_{1}|^{2}dη_{x_{0}}(x_{1}))^{1/2} + f^{n}(x_{0})
≤ 2g(x_{0}).
□

Proposition 4.32. For any γ ∈ 𝒫_{2}(ℝ^{2d})_{μ} ⊂ 𝒫_{2}(ℝ_{0}^{d} × ℝ_{1}^{d}) there is a unique optimal 3-plan α_{γ}^{P(γ)} in OPT_{μ}(γ, P(γ)). Moreover this plan is induced by a map in the following sense: there exists a Borel map P_{γ} : ℝ_{0}^{d} × ℝ_{1}^{d} → ℝ_{2}^{d} such that

(4.36)  α_{γ}^{P(γ)} = (π^{0}, π^{1}, P_{γ} ∘ (π^{0}, π^{1}))_{#}γ.

Observe that condition (4.36) may be given in the following equivalent way: for μa.e. x_{0} the plan (α_{γ}^{P(γ)})_{x_{0}} ∈ Opt(γ_{x_{0}}, (P(γ))_{x_{0}}) is induced by the map x_{1} → P_{γ}(x_{0}, x_{1}).

Proof. To prove the thesis is sufficient to show that any optimal 3-plan α ∈ OPT_{μ}(γ, P(γ)) is induced by a map: indeed if T_{1}, T_{2} : ℝ_{0}^{d} × ℝ_{1}^{d} → ℝ_{2}^{d} were two different optimal maps, then the plan

\frac{1}{2}((π^{0}, π^{1}, T_{1} ∘ (π^{0}, π^{1}))_{#}γ + (π^{0}, π^{1}, T_{2} ∘ (π^{0}, π^{1}))_{#}γ),

would be optimal and not induced by a map.

So fix α ∈ OPT_{μ}(γ, P(γ)), let dα = dγ ⊗ dα_{x_{0},x_{1}} be its disintegration it w.r.t. γ and define

T (x_{0}, x_{1}) := ∫ x_{2}dα_{x_{0},x_{1}}(x_{2}),

and \overline{γ} := (π^{0}, T ∘ (π^{0}, π^{1}))_{#}γ. Since clearly T (x_{0}, x_{1}) ∈ \overline{Conv} (supp(α_{x_{0},x_{1}})) for γ-a.e. (x_{0}, x_{1}), and

(P(γ))_{x_{0}} = ∫ α_{x_{0},x_{1}}d(π_{#}^{1}α)(x_{1}),

we get

supp ((\overline{γ})_{x_{0}}) ⊂ \overline{Conv} ( supp ((P(γ))_{x_{0}})).

Applying the previous lemma we get \overline{γ} ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})), moreover the plan (π^{0}, π^{1}, T ∘ (π^{0}, π^{1}))_{#}γ is admissible for the couple γ, \overline{γ}, and therefore

\begin{matrix}W_{μ}^{2}(γ, \overline{γ}) ≤ & ∫ |x_{1} − T (x_{0}, x_{1})|^{2}dγ(x_{0}, x_{1}) \\ = & ∫ |∫ (x_{1} − x_{2})dα_{x_{0},x_{1}}(x_{2})|^{2} dγ(x_{0}, x_{1}) \\ ≤ & ∫ |x_{1} − x_{2}|^{2}dα_{x_{0},x_{1}}(x_{2})dγ(x_{0}, x_{1}) = W_{μ}^{2}(γ, P(γ)). \end{matrix}

Since the second inequality is strict if α_{x_{0},x_{1}} is not a Dirac mass for γ-a.e. (x_{0}, x_{1}), by the minimizing properties of P(γ) we get the validity of equation (4.36). □

The first remarkable properties of the plan α_{γ}^{P(γ)} are collected below.

Theorem 4.33. Let γ ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) be any plan. Then, for every 4-plan β satisfying

(4.37a)  π_{#}^{0,1,2}β = α_{γ}^{P(γ)},
(4.37b)  π_{#}^{0,3}β = η ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})),

it holds:

i) ⟨x_{1}, x_{3}⟩_{β} = ⟨x_{2}, x_{3}⟩_{β}.
ii) ⟨x_{1}, x_{3}⟩_{β} = ⟨γ, η⟩ if and only if ⟨x_{2}, x_{3}⟩_{β} = ⟨P(γ), η⟩.
iii) π_{#}^{0,1,3}β ∈ OPT_{μ}(γ, η) if and only if π_{#}^{0,2,3}β ∈ OPT_{μ}(P(γ), η).

Conversely let \overline{γ} be a plan in Tan_{μ}(𝒫_{2}(ℝ^{d})), and assume that there exists a 3-plan α ∈ ADM_{μ}(γ, \overline{γ}) such that for every 4-plan β satisfying

(4.38a)  π_{#}^{0,1,2}β = α,
(4.38b)  π_{#}^{0,3}β ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})),

it holds

⟨x_{1}, x_{3}⟩_{β} = ⟨x_{2}, x_{3}⟩_{β}.

Then \overline{γ} = P(γ) and α = α_{γ}^{P(γ)}.

Proof. It is clear that ii) and iii) follow directly from i). To prove i) observe that γ_{ε} := (π^{0}, π^{2} + επ^{3})_{#}β ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) for any ε ∈ ℝ, and evaluate

W_{μ}^{2}(γ, γ_{ε}) ≤ ∥x_{1} − x_{2} − εx_{3}∥_{β}^{2} = ∥x_{1} − x_{2}∥_{β}^{2} − 2ε⟨x_{1} − x_{2}, x_{3}⟩_{β} + ε^{2}∥x_{3}∥_{β}.

To get the thesis let ε tend to 0^{+} and 0^{−} and use the minimality of γ_{0} = P(γ). Let us turn to the converse implication: choose any 3-plan α such that π_{#}^{0,1}α =

γ, \overline{γ} := π_{#}^{0,2}α ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) and assume that α ≠ α_{γ}^{P(γ)}. We will show that there exists a 4-plan β satisfying (4.38), for which ⟨x_{1} − x_{2}, x_{3}⟩β is different from 0. Indeed for any \overline{β} such that

π_{#}^{0,1,2}\overline{β} = α,
π_{#}^{0,1,3}\overline{β} = α_{γ}^{P(γ)},

we have that (thanks to what we just proved, and the fact that (π^{0}, π^{2} − π^{3})_{#}β ∈ Tan_{μ}(𝒫_{2}(ℝ^{d}))) ⟨x_{1} − x_{3}, x_{2} − x_{3}⟩\overline{β} = 0. Therefore we can conclude with

\begin{matrix}0 ≠ ∥x_{2} − x_{3}∥2 ; \overline{β} = & ⟨x_{2} − x_{3}, x_{2} − x_{1}⟩\overline{β} + ⟨x_{2} − x_{3}, x_{1} − x_{3}⟩\overline{β} \\ = & ⟨x_{2} − x_{3}, x_{2} − x_{1}⟩\overline{β} = ⟨x_{3}, x_{2} − x_{1}⟩_{β}, \end{matrix}

where β := (π^{0}, π^{1}, π^{2}, π^{2} − π^{3})_{#}\overline{β} satisfies (4.38). □

Corollary 4.34. Let γ ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) be any plan. The plan P(γ) is the unique element of Tan_{μ}(𝒫_{2}(ℝ^{d})) with the following property:

(4.39)  ⟨γ, η⟩ = ⟨P(γ), η⟩, ∀η ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})).

Proof. From i) and ii) of the previous theorem it follows easily that P(γ) has the stated property. Now suppose that σ ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) satisfies (4.39), too, and observe that from

∥P(γ)∥^{2} = ⟨P(γ), γ⟩ = ⟨P(γ), σ⟩ = ⟨γ, σ⟩ = ∥σ∥^{2},

we obtain ∥P(γ)∥ = ∥σ∥ and ⟨σ, P(γ)⟩ = ∥P(γ)∥^{2} = ∥σ∥^{2}, from which the thesis follows. □

The theorem we just proved allows us to prove the “linearity” of P stated in the following theorem.

Theorem 4.35 (Linearity of P). Let γ_{1}, γ_{2} be plan in 𝒫_{2}(ℝ^{2d})_{μ}, and let λ be a real number. Then it holds:

(4.40a)  P(λ · γ_{1}) = λ · P(γ_{1}),
(4.40b)  P(γ_{1} ⊕ γ_{2}) = P(γ_{1}) ⊕ P(γ_{2}).

Proof. The first equation follows easily by the observation that for every couple of plan γ, η it holds: α ∈ ADM_{μ}(γ, η) if and only if λ · α ∈ ADM_{μ}(λ · γ, λ · η), where λ · α is intended to be the plan (π^{0}, λπ^{1}, λπ^{2})_{#}α.

Now we turn to the second one. Fix a 3-plan α admissible for the couple γ_{1}, γ_{2} and define γ_{3} := γ_{1} +_{α} γ_{2} = (π^{0}, π^{1} + π^{2})_{#}α. Our goal is to prove that there exists a 3plan which we call P(α) admissible for the couple P(γ_{1}), P(γ_{2}), such that P(γ_{1})+_{P(α)} P(γ_{2}) = P(γ_{3}). Let us use the variables x_{1}, x_{2}, x_{3}, x_{4} for the plans γ_{1}, γ_{2}, P(γ_{1}), P(γ_{2}), respectively. Choose a 5-plan β satisfying

(4.41a)  π_{#}^{0,1,2}β = α,
(4.41b)  π_{#}^{0,1,3}β = α_{γ_{1}}^{P(γ_{1})},
(4.41c)  π_{#}^{0,2,4}β = α_{γ_{2}}^{P(γ_{2})},

and define P(α) := π_{#}^{0,2,4}β (observe that since α_{γ_{i}}^{P(γ_{i})} is induced by a map on each fibre - see proposition 4.32 - there is a unique β satifying the above conditions, so P(α) is actually a well defined function of α). With this notation our claim becomes: prove that the projetion of (π^{0}, π^{1} + π^{2})_{#}β is (π^{0}, π^{3} + π^{4})_{#}β. In order to prove this we will use the second part of the theorem 4.33: choose any 6-plan B satisfying:

π_{#}^{0,1,2,3,4}B = β,
π_{#}^{0,5}B ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})).

All we have to prove is the equality

⟨x_{1} + x_{2}, x_{5}⟩_{B} = ⟨x_{3} + x_{4}, x_{5}⟩_{B},

whose proof is obvious, since ⟨x_{1}, x_{5}⟩_{B} = ⟨x_{3}, x_{5}⟩_{B} and ⟨x_{2}, x_{4}⟩_{B} = ⟨x_{3}, x_{5}⟩_{B} because of theorem 4.33 applied to the couples γ_{1}, P(γ_{1}) and γ_{2}, P(γ_{2}). □

Observe that in particular we proved that

P(γ_{1} +α γ_{2}) = P(γ_{1}) +_{P(α)} P(γ_{2}).

Remark 4.36. It is not clear to the author whether the plan P(α) is optimal whenever α is.

In the following proposition we show that the map P is 1-Lipschitz.

Hilbert case 2. Let V ⊂ H be a closed subspace of an Hilbert space H, let v_{1}, v_{2} ∈ H be two fixed vectors and let P : H → V be the orthogonal projection. Then |P (v_{1}) − P (v_{2})| ≤ |v_{1} − v_{2}|.

Proof. We have that

\begin{aligned}⟨v_{1} − P (v_{1}), P (v_{1}) − P (v_{2})⟩ & = 0, \\ ⟨v_{2} − P (v_{2}), P (v_{1}) − P (v_{2})⟩ & = 0. \end{aligned}

Therefore the thesis follows from

|v_{1} − v_{2}|^{2} =|P (v_{1}) − P (v_{2}) + (v_{1} − P (v_{1})) + (v_{2} − P (v_{2}))|^{2}
=|P (v_{1}) − P (v_{2})|^{2} + 2⟨P (v_{1}) − P (v_{2}), v_{1} − P (v_{1})⟩
+ 2⟨P (v_{1}) − P (v_{2}), v_{2} − P (v_{2})⟩ + |v_{1} − P (v_{1}) + v_{2} − P (v_{2})|^{2}
=|P (v_{1}) − P (v_{2})|^{2} + |v_{1} − P (v_{1}) + v_{2} − P (v_{2})|^{2} ≥ |P (v_{1}) − P (v_{2})|^{2}.
□

Corollary 4.37 (Non expansivity of P). Let γ_{1}, γ_{2} ∈ 𝒫_{2}(ℝ^{2d})_{μ} be any couple of plans. Then it holds

W_{μ}(P(γ_{1}), P(γ_{2})) ≤ W_{μ}(γ_{1}, γ_{2}).

Proof. Let x_{1}, x_{2}, x_{3}, x_{4} be the variables of γ_{1}, γ_{2}, P(γ_{1}), P(γ_{2}). Choose α ∈ OPT_{μ}(γ_{1}, γ_{2}) and find a 5-plan β satisfying

\begin{aligned}π_{#}^{0,1,2}β & = α, \\ π_{#}^{0,1,3}β & = α_{γ_{1}}^{P(γ_{1})}, \\ π_{#}^{0,2,4}β & = α_{γ_{2}}^{P(γ_{2})}. \end{aligned}

Since (π^{0}, π^{3} − π^{4})_{#}β ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) we know from theorem 4.33 that it holds

\begin{aligned}⟨x_{3} − x_{4}, x_{1} − x_{3}⟩_{β} & = 0, \\ ⟨x_{3} − x_{4}, x_{4} − x_{2}⟩_{β} & = 0, \end{aligned}

therefore we conclude with:

W_{μ}^{2}(γ_{1}, γ_{2}) =∥x_{1} − x_{2}∥_{β}^{2} = ∥(x_{3} − x_{4}) + (x_{1} − x_{3}) + (x_{4} − x_{2})∥_{β}^{2}
=∥x_{3} − x_{4}∥_{β}^{2} + 2⟨x_{3} − x_{4}, x_{1} − x_{3}⟩_{β}
+ 2⟨x_{3} − x_{4}, x_{4} − x_{2}⟩_{β} + ∥x_{1} − x_{3} + x_{4} − x_{2}∥_{β}^{2}
=∥x_{3} − x_{4}∥_{β}^{2} + ∥x_{1} − x_{3} + x_{4} − x_{2}∥_{β}^{2} ≥ W_{μ}(P(γ_{1}), P(γ_{2})).
□

Observe that from this corollary it follows in particular (taking γ_{2} = 0_{μ}) that

∥P(γ)∥ ≤ ∥γ∥.

We now analyze the properties of P w.r.t. some of the other functions we defined. Recall that P_{μ} is the orthogonal projection of L_{μ}^{2} onto Tan_{μ}(𝒫_{2}(ℝ^{d})), that ℑ(f) = (Id, f)_{#}μ and that ℬ is the barycentric projection.

Proposition 4.38 (Further properties of P). It holds

P(P(γ)) = P(γ),
⟨P(γ_{1}), γ_{2}⟩ = ⟨γ_{1}, P(γ_{2})⟩,
P(ℑ(f)) = ℑ(P_{μ}(f)),
P_{μ}(ℬ(γ)) = ℬ(P(γ)).

Proof. The first identity is obvious. For the second one observe ⟨P(γ_{1}), γ_{2}⟩ = ⟨P(γ_{1}), P(γ_{2})⟩ = ⟨γ_{1}, P(γ_{2})⟩. To prove the third one observe that it is sufficient to show that the projection of a plan induced by a map is induced by a map, too: to prove this it is sufficient to recall that the plan α_{γ}^{P(γ)} is induced by a map in the sense of equation (4.36). To prove the last equation recall the properties of the objects involved:

\overline{γ} = P(γ) ⇔ \overline{γ} ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})), and ⟨γ, η⟩ = ⟨\overline{γ}, η⟩, ∀η ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})),
f = ℬ(γ) ⇔ ⟨γ, ℑ(g)⟩ = ⟨f, g⟩_{μ},  ∀g ∈ L_{μ}^{2},
v = P_{μ}(f) ⇔ v ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) and ⟨f, g⟩_{μ} = ⟨v, g⟩_{μ}, ∀g ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})).

Therefore v = P_{μ}(ℬ(γ))

⇔ v ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) and ⟨ℬ(γ), g⟩_{μ} = ⟨v, g⟩_{μ} ∀g ∈ Tan_{μ}(𝒫_{2}(ℝ^{d}))
⇔ v ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) and ⟨γ, ℑ(g)⟩ = ⟨v, g⟩_{μ} ∀g ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})),

and on the other hand \overline{v} = ℬ(P(γ))

⇔ ⟨P(γ), ℑ(f)⟩ = ⟨v, f⟩_{μ} ∀f ∈ L_{μ}^{2},
⇔ ⟨γ, ℑ(P_{μ}(f))⟩ = ⟨v, f⟩_{μ} ∀f ∈ L_{μ}^{2},

therefore clearly v = \overline{v}. □

Before proving our final result, which regards the infinitesimal behavior of the exponential map, we state a lemma regarding convergence of P, which, although not used in the sequel, is of its own interest. We already showed that P is Lipschitz on (𝒫_{2}(ℝ^{2d})_{μ}, W_{μ}), but what about the weak continuity? The following example shows that in general we cannot hope for weak continuity.

Example 4.39. Let μ := (2π)^{−1}L^{1}|_{[0,2π]}, v_{n}(x) := sin nx. It is not difficult to

see that, given that d = 1, it holds Tan_{μ}(𝒫_{2}(ℝ^{d})) = L_{μ}^{2}. In particular we have that v_{n} ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) which implies P(ℑ(v_{n})) = ℑ(v_{n}). Let γ be the W −limit of (ℑ(v_{n})):

dγ(x_{0}, x_{1}) = \frac{1}{(2π^{2})\sqrt{1 − x_{1}^{2}}} d(x_{0}, x_{1})|_{Q},

where Q = [0, 2π] × [−1, 1]. In particular γ is not induced by a map. Given that μ is absolutely continuous, the space Tan_{μ}(𝒫_{2}(ℝ^{d})) is isometric to Tan_{μ}(𝒫_{2}(ℝ^{d})) through the map ℑ. It follows that γ ∉ Tan_{μ}(𝒫_{2}(ℝ^{d})) and therefore P(ℑ(v_{n})) = ℑ(v_{n}) can’t converge (not even weakly) to P(γ) since the W −limit of the sequence is γ and does not belong to Tan_{μ}(𝒫_{2}(ℝ^{d})).

Lemma 4.40. Let (γ_{n}) ⊂ 𝒫_{2}(ℝ^{2d})_{μ} be a sequence of plans τ−converging to some plan γ. Then for every τ−accumulation point σ of P(γ_{n}) it holds P(γ) = P(σ). In particular

∥P(γ)∥ ≤ \underline{lim}_{n→∞} ∥P(γ_{n})∥.

Proof. We can assume without loss of generality that α_{γ_{n}}^{P(γ_{n})} τ−converge to some α with π_{#}^{0,2}α = σ.

We argue by contradiction: suppose that P(γ) ≠ P(σ). Choose a 5-plan B satisfying

\begin{aligned}π_{#}^{0,1,2}B & = α, \\ π_{#}^{0,1,3}B & = α_{γ}^{P(γ)}, \\ π_{#}^{0,2,4}B & = α_{σ}^{P(σ)}, \end{aligned}

and use part i) of the theorem 4.33 (observe that (π^{0}, π^{3} − π^{4})_{#}B ∈ Tan_{μ}(𝒫_{2}(ℝ^{d}))) to get

\begin{matrix}0 < W_{μ}^{2}(P(γ), P(σ)) ≤ & ∥x_{3} − x_{4}∥_{B}^{2} \\ = & ⟨x_{3}, x_{3} − x_{4}⟩_{B} − ⟨x_{4}, x_{3} − x_{4}⟩_{B} \\ = & ⟨x_{1}, x_{3} − x_{4}⟩_{B} − ⟨x_{2}, x_{3} − x_{4}⟩_{B} \\ = & ⟨x_{1} − x_{2}, x_{3} − x_{4}⟩_{B}. \end{matrix}

Define the plan β := (π^{0}, π^{1}, π^{2}, π^{3} − π^{4})_{#}B and observe that we just proved

(4.43)  ⟨x_{1} − x_{2}, x_{3}⟩_{β} > 0.

We want to find a sequence β_{n} τ−converging to β satisfying

\begin{aligned}π_{#}^{0,1,2}β_{n} & = α_{γ_{n}}^{P(γ_{n})}, \\ π_{#}^{0,3}β_{n} & = η. \end{aligned} ; (4.44)

In order to do so, find, for every n ∈ ℕ, a 7-plan B_{n} satisfying

\begin{aligned}π_{#}^{0,1,2,3}B_{n} & = β, \\ π_{#}^{0,1,2,4,5,6}B_{n} & ∈ Opt(α_{γ}^{P(γ)}, α_{γ_{n}}^{P(γ_{n})}), \end{aligned}

where in the second equation we thought α_{γ}^{P(γ)} ∈ 𝒫_{2}(ℝ_{0}^{d} × ℝ_{1}^{d} × ℝ_{2}^{d}) and α_{γ}^{P(γ)} ∈ 𝒫_{2}(ℝ_{4}^{d} × ℝ_{5}^{d} × ℝ_{6}^{d}). The τ−convergence of α_{γ_{n}}^{P(γ_{n})} to α_{γ}^{P(γ)} implies that (B_{n}) τ−converges to (π^{0}, π^{1}, π^{2}, π^{3}, π^{0}, π^{1}, π^{2})_{#}β. The 4-plans β_{n} := (π^{4}, π^{5}, π^{6}, π^{3})_{#}B_{n} satisfy (4.44), therefore from part i) of theorem 4.33 we have

⟨x_{5} − x_{6}, x_{3}⟩_{B_{n}} = ⟨x_{5} − x_{6}, x_{3}⟩_{β_{n}} = 0.

Passing to the limit as n → ∞ we get

0 = lim_{n→∞} ⟨x_{6} − x_{5}, x_{3}⟩_{B_{n}} = ⟨x_{1} − x_{2}, x_{3}⟩_{β},

which contradicts (4.43). Therefore it must hold P(γ) = P(σ).

For the second statement observe that the τ−convergence of α_{γ_{n}}^{P(γ_{n})} to α implies the τ−convergence of P(γ_{n}) to σ, therefore:

∥P(γ)∥ = ∥P(σ)∥ ≤ ∥σ∥ ≤ \underline{lim}_{n→∞} ∥P(γ_{n})∥.

□

The next (and final) theorem regards the infinitesimal behavior of the exponential function: there we prove that the two curves t → exp(t·γ) and t → exp(t·P(γ)) behave in the same way near 0 under the assumption that the plan γ is induced by a map. For the general case we are able only to prove only a partial result.

Theorem 4.41 (Infinitesimal behavior of exp_{μ}). Let γ ∈ 𝒫_{2}(ℝ^{2d})_{μ} be a plan, μ_{t} := exp(t · γ) (so that μ_{0} = μ). Let η_{t} be any choice of optimal plans between μ_{0} and μ_{t} (i.e. η_{t} ∈ Opt(μ_{0}, μ_{t})), t ∈ [0, 1], and define

σ_{t} := (π^{0}, \frac{π^{1} − π^{0}}{t} )_{#} η_{t}.

Then it holds

(4.45)  \overline{lim}_{t→0^{+}} ⟨σ_{t}, σ⟩ ≤ ⟨P(γ), σ⟩, ∀σ ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})).

Proof. By density and positive 1-homogeneity it is sufficient to prove equation (4.45) for all the plans σ ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) such that (π^{1}, π^{1} + π^{2})_{#}σ is the unique optimal plan between its marginals. With this assumption let us evaluate \frac{d}{dt}W^{2}(μ_{t}, σ) at t = 0, where σ := (π^{1} + π^{2})_{#}σ. From proposition 4.10 we know that it is equal to −2⟨γ, σ⟩ = −2⟨P(γ), σ⟩, and on the other hand taking any α_{t} ∈ OPT_{μ}(η_{t}, (π^{1}, π^{1} + π^{2})_{#}σ) we have

\begin{matrix}\frac{d}{dt}|_{t=0}W^{2}(μ_{t}, μ_{0}) = & lim_{t→0^{+}} \frac{W^{2}(μ_{t}, ν) − W^{2}(μ_{0}, ν)}{t} \\ ≤ & lim_{t→0^{+}} \frac{∥x_{1} − x_{2}∥_{α_{t}}^{2} − ∥x_{0} − x_{2}∥_{α_{t}}^{2}}{t} \\ = & − 2⟨\frac{x_{1} − x_{0}}{t} , x_{2} − x_{0}⟩_{α_{t}} = ⟨σ_{t}, σ⟩. \end{matrix}

□

It may seem curious that for general plan we obtain only an inequality in the limit of scalar products, instead of an equality, however (although the author doesn’t know whether the equality holds or not) we want to point out why it is not possible to derive the equality from the inequality. More explicitely: for every μ and every γ ∈ 𝒫_{2}(ℝ^{2d})_{μ} there exists γ′ ∈ 𝒫_{2}(ℝ^{2d})_{μ} such that ⟨γ′, σ⟩ ≥ ⟨γ, σ⟩ for any σ ∈ 𝒫_{2}(ℝ^{2d})_{μ}, with strict inequality for some σ. To see this, choose any vector v ∈ ℝ^{d} different from 0, define the functions f^{±}(x_{0}, x_{1}) := (x_{0}, x_{1} ± v) and the plan γ′ := 1/2(f_{#}^{+}γ + f_{#}^{−}γ). Then it is easy to see that for every σ ∈ 𝒫_{2}(ℝ^{2d})_{μ} and every 3-plan α ∈ OPT_{μ}(γ, σ), there exists α′ ∈ ADM_{μ}(γ′, σ) such that ⟨x_{1}, x_{2}⟩_{α} = ⟨x_{1}, x_{2}⟩_{α_{′}}: define α′ := 1/2(\overline{f}_{#}^{+}α + \overline{f}_{#}^{−}α), where \overline{f}^{±}(x_{0}, x_{1}, x_{2}) := (x_{0}, x_{1} ± v, x_{2}). This shows that the inequality ⟨γ′, σ⟩ ≥ ⟨γ, σ⟩ always holds. To show that sometimes it has to be strict, observe that ∥γ′∥ > ∥γ∥, therefore at least one of the two inequalities in ∥γ′∥^{2} ≥ ⟨γ′, γ⟩ ≥ ∥γ∥^{2} has to be strict.

We conclude the chapter by underlying an interesting fact regarding curves of the type t → exp(t · γ), which actually creates difficulties when trying to generalize the previous theorem to general plans and shows that the parallel transport does not always exist (see chapter 6).

Fix any plan γ ∈ 𝒫_{2}(ℝ^{2d})_{μ} and consider the curve μ_{t} := exp(t · γ). This curve is clearly absolutely continuous, therefore considering its tangent vector field v_{t} (which exist for a.e. t), we know from proposition 3.9 that for every ν ∈ 𝒫_{2}(ℝ^{d}) and a.e. t it holds

\frac{d}{dt^{+}} W^{2}(μ_{t}, ν) = −2⟨v_{t}(x), ℬ(η_{t}) − Id⟩_{μ_{t}},

where η_{t} is any plan in Opt(μ_{t}, ν). On the other hand proposition 4.10 ensures that the same derivative is equal to −2⟨γ_{t}, (π^{0}, π^{1} − π^{0})_{#}η_{t}⟩, where γ_{t} := (π^{0} + tπ^{1}, π^{1})_{#}γ. The characterization we did of the projection of a plan implies that for a.e. t it holds

(4.46)  P(γ_{t}) = v_{t}.

In order to prove the previous theorem for general plans we would like to write the previous equation for t = 0, however this cannot be done, because we don’t know whether v_{t} is well defined at t = 0.

The interesting fact is that the metric derivative |μ̇_{t}| can exists for every t and be discontinuous at 0. Given that for a.e. t it holds |μ̇_{t}| = ∥v_{t}∥_{μ_{t}} = ∥P(γ_{t})∥, this partially explains why it is difficult to find properties of P(γ) by taking the limit as t → 0 of similar properties of P(γ_{t}).

The following is an example of a plan γ which is induced by a map for which |μ̇_{0}| ≤ 1, while |μ̇_{t}| is costant and bigger than 1 for positive t; moreover for positive t the curve μ_{t} is locally a geodesic (but is not a geodesic in any interval [0, T]) and the plans (actually functions) γ_{t} are tangent in μ_{t}.

Example 4.42. Let Q = [0, 1]×[0, 1] be the unit square in ℝ^{2} and T_{i}, i = 1, 2, 3, 4 be the four triangles in which Q is divided by its diagonal. Define the function f : Q → ℝ^{2} as

\begin{aligned}f|_{T_{1}} & := (1, 1), \\ f|_{T_{2}} & := (−1, 1), \\ f|_{T_{3}} & := (−1, −1), \\ f|_{T_{4}} & := (1, −1). \end{aligned}

Now define μ := L^{2}|_{Q} and observe that f ∈ L_{μ}^{2} and that for t > 0, μ_{t} := (Id + tf)_{#}μ is made of 4 connected components, each one the traslation of one of the T_{i}.

The function f_{t} := f ∘ (Id + tf)^{−1} is constant on each of those components and therefore clearly tangent. It follows that for positive t we have |μ̇_{t}| = ∥f_{t}∥_{μ_{t}} = ∥f∥_{μ} = \sqrt{2}. Our example will be concluded if we show that |μ̇_{0}| ≤ 1. To see this consider the function g on Q defined as

\begin{aligned}g|_{T_{1}} & := (0, 1), \\ g|_{T_{2}} & := (−1, 0), \\ g|_{T_{3}} & := (0, −1), \\ g|_{T_{4}} & := (1, 0), \end{aligned}

and let ν_{t} := (Id + tg)_{#}μ.

Then clearly W (μ_{t}, ν_{t}) ≤ ∥f − g∥_{μ} = 1, therefore the conclusion will follow if we show that W (ν_{t}, μ) = o(t). This can be proved either by applying theorem 4.41 and checking that P_{μ}(g) = 0, or, equivalently but in a more geometrical way, by finding a family of transport maps whose cost goes to 0 faster than t: for the latter approach see the picture below (a mass of order t is moved of a distance of order t, giving a cost of order t^{3/2}).

72 4. THE GEOMETRIC TANGENT SPACE

CHAPTER 5 Geodesically convex sets

The aim of this chapter is to study the τ−closure properties of geodesically convex sets. The reasons of such an interest come from the fact that the distance W was recently studied because of the strict relations with some evolution PDE’s which may be interpreted as curves of maximal slope of certain geodesically convex functionals, i.e. functionals that are convex along geodesics. Such an approach, introduced by Otto in [46] and then further analyzed by Carrillo-McCann-Villani in [25], by Agueh in [2] and by the author together with Ambrosio and Savaré in [11] (see [11] for more detailed references), leads to the study of the problem of existence and uniqueness of those curves: the theory of minimizing movements introduced by De Giorgi ([29]) provides a satisfactory answer to these questions under only weak compactness assumptions. In [11] there are mainly two theorems on existence of curves of maximal slope for geodesically convex functionals which rely on two different kind of assumptions on the functional F:

i) F is lower semicontinuous w.r.t. the weak topology and satisfies F (γ[t]) ≤ (1 − t)F (γ[0]) + tF (γ[1]) for any optimal plan γ ∈ 𝒫_{2}(ℝ^{2d}) (see Corollaries 2.4.11 and 2.4.12 of [11]),

ii) F is lower semicontinuous w.r.t. the strong topology and satisfies F (γ[t]) ≤ (1 − t)F (γ[0]) + tF (γ[1]) for any plan γ ∈ 𝒫_{2}(ℝ^{2d}) (see Theorem 4.0.4 of [11]).

The two notions of convexity along geodesics just introduced are strictly related to the following notions of geodesic convexity for sets:

Definition 5.1 (Geodesically convex sets). We will say that a set C ⊂ 𝒫_{2}(ℝ^{d}) is geodesically convex if for any μ_{1}, μ_{0} ∈ C there exists a γ ∈ Opt(μ_{0}, μ_{1}) such that the whole segment joining μ_{0} to μ_{1} through γ belongs to C, that is:

[γ](t) ∈ C, ∀t ∈ [0, 1].

Definition 5.2 (Strongly geodesically convex sets). We will say that a set C ⊂ 𝒫_{2}(ℝ^{d}) is strongly geodesically convex if for any μ_{1}, μ_{0} ∈ C and every γ ∈ Adm(μ_{0}, μ_{1}) the whole segment joining μ_{0} to μ_{1} through γ belongs to C, that is:

[γ](t) ∈ C, ∀t ∈ [0, 1].

It is easy to check that a functional F l.s.c. w.r.t. the strong topology is convex along geodesics in the sense of (i) (respectively, (ii)), if and only if its sublevels are geodesically convex (respectively, strongly geodesically convex).

The main result of this chapter is to show that in case (i) the assumption of lower semicontinuity w.r.t. τ is redundant and may be substituted with semicontinuity w.r.t. W, provided we know that F attains the value +∞ on non regular measures. In order to prove this we will show that any W −closed geodesically convex subset of 𝒫_{2}(ℝ^{d}) is τ−closed.

The chapter has the following structure: we first introduce the concept of halfspace and study its basic properties, after that we prove that any W −closed strongly geodesically convex set is τ−closed (thus in consistency with the results proven under the first kind of hypothesis on the functional F), and finally we turn to W −closed geodesically convex subsets of 𝒫_{2}(ℝ^{d}) and we prove that those sets are τ−closed, too. In the last section we will shortly discuss the definition of the subdifferential of a geodesically convex functional and we analyze the differentiability properties of W^{2}(·, σ).

As in the last sections of the previous chapter it will happen that a theorem stated in the Wasserstein space is the analogous of a theorem valid in Hilbert spaces, in this case we will anticipate the proof of our theorem with the simpler one valid in a Euclidean setting.

1. Halfspaces

In this section we define the halfspaces and discuss their basic properties.

Definition 5.3 (Halfspace). Let γ ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) be any tangent plan, and C ∈ ℝ a constant. The two halfspaces identified by γ and C are

ℋ_{γ;C}^{+} := {ν ∈ 𝒫_{2}(ℝ^{d}) : ⟨η, γ⟩ ≥ C, for some η ∈ exp_{μ}^{−1}(ν)} ,
ℋ_{γ;C}^{−} := {ν ∈ 𝒫_{2}(ℝ^{d}) : ⟨η, γ⟩ ≤ C, for some η ∈ exp_{μ}^{−1}(ν)} .

For regular measures μ and v ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) we will write ℋ_{v;C}^{±}, in this case the definition simplifies in

\begin{aligned}ℋ_{v;C}^{+} & := \{ν : ⟨T_{μ}^{ν} − Id, v⟩_{μ} ≥ C\} , \\ ℋ_{v;C}^{−} & := \{ν : ⟨T_{μ}^{ν} − Id, v⟩_{μ} ≤ C\} . \end{aligned}

Proposition 5.4. Let γ ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) be a plan and C ∈ ℝ a real number. Then it holds:

i) The halfspace ℋ_{γ;C}^{+} is weakly closed (i.e. closed w.r.t. τ).

ii) If the plan γ is induced by a map (and in particular if the base measure μ is regular), then the halfspace ℋ_{γ;C}^{−} is weakly closed, too.

Proof. Let us start with i): consider a sequence ν_{n} τ−converging to ν and belonging to ℋ_{γ;C}^{+}, and a sequence (η_{n}) such that η_{n} ∈ exp_{μ}^{−1}(ν_{n}) such that ⟨γ, η_{n}⟩ ≥ C, for all n. Then choose α_{n} ∈ OPT_{μ}(γ, η_{n}) and observe that there must exist a subsequence α_{n_{k}} which τ−converges to some α such that η := π_{#}^{0,2}α ∈ exp_{μ}^{−1}(ν). Corollary 2.29 ensures that

⟨γ, η⟩ ≥ ⟨x_{1}, x_{2}⟩_{α} = lim_{k} ⟨x_{1}, x_{2}⟩_{α_{n_{k}}} ≥ C.

The claim ii) follows with the same idea. Use the same notation as before and assume that ⟨γ, η_{n}⟩ ≤ C. Observe that if γ = (Id, T)_{#}μ, for any plan \overline{γ} ∈ 𝒫_{2}(ℝ^{2d})_{μ} the set OPT_{μ}(γ, \overline{γ}) contains only one element, namely (π^{0}, T ∘ π^{0}, π^{1})_{#}\overline{γ}. Therefore the limit plan α is optimal and it holds

⟨γ, η⟩ = ⟨x_{1}, x_{2}⟩_{α} = lim_{k} ⟨x_{1}, x_{2}⟩_{α_{n_{k}}} ≤ C.

□

Remark 5.5. It is important to observe that, for non regular measure μ, an halfspace ℋ_{γ;C}^{−} may fail to be τ−closed. Moreorer, in general this set may fail even to be W −closed. Indeed consider the following example: let μ ∈ 𝒫(ℝ^{2}) be defined as the 1-dimensional Hausdorff measure restricted to the interval [0, 1] × {0}, consider the subsets A_{n} of [0, 1] defined as

A_{n} := ⋃_{i=0}^{2^{n}−1} [\frac{2i}{2^{n+1}} , \frac{2i + 1}{2^{n+1}}] ,

and define the functions f_{n} : ℝ → ℝ as

f_{n} := \{ \begin{matrix}1 & if x ∈ A_{n}, \\ −1 & if x ∉ A_{n}. \end{matrix}

Then it is not difficult to see that if we define ν_{n} := (Id, f_{n})_{#}μ (where we think μ as a measure on ℝ), then the plans γ_{n} induced by the maps (Id, f_{n}) are the only optimal plans between μ an ν_{n}. Moreover it is clear that the v_{n} W -converge to the measure

ν := \frac{1}{2} ((Id, 1)_{#}μ + (Id, −1)_{#}μ) ,

having identified once again μ with its restriction to ℝ, and that there is only one optimal plan between μ and ν, namely the plan γ whose disintegration w.r.t. to dμ(x, y) is given by the family 1/2(δ_{x,1} + δ_{x,−1}). Then, we have that W_{μ}(γ_{n}, γ) = 1 for every n, and that ∥γ_{n}∥ = ∥γ∥ = 1, therefore from the equality

W_{μ}^{2}(γ_{n}, γ) = ∥γ∥^{2} + ∥γ_{n}∥^{2} − 2⟨γ, γ_{n}⟩

we gain lim ⟨γ, γ_{n}⟩ = 1/2. As a consequence, for any C satisfying 1/2 < C < 1 the halfspace ℋ_{γ;C}^{−} is not even W −closed, since ν_{n} ∈ ℋ_{γ;C}^{−} for every n, but ν ∉ ℋ_{γ;C}^{−}.

Remark 5.6. The definition of halfspace make sense for any γ ∈ 𝒫_{2}(ℝ^{2d})_{μ}, and not only for those in Tan_{μ}(𝒫_{2}(ℝ^{d})). We restricted the interest to tangent plans because of corollary 4.34, which ensures that ℋ_{γ;C}^{±} = ℋ_{P(γ);C}^{±}, where P(γ) is the projection of γ onto Tan_{μ}(𝒫_{2}(ℝ^{d})).

2. Strongly geodesically convex sets

In this section we assume that the W −closed set C satisfies the strong geodesic convexity hypothesis given in definition 5.2, and we will prove that it is weakly closed.

The first step of the proof is to prove that for any point μ ∈ 𝒫_{2}(ℝ^{d}) there exists a point in C of minimal distance from μ (the proof is very similar to that of proposition 4.30).

Proposition 5.7. Let C ⊂ 𝒫_{2}(ℝ^{d}) be a W −closed strongly geodesically convex set and let μ ∈ 𝒫_{2}(ℝ^{d}) be a measure. Then there exists a measure ν ∈ C which minimizes the distance from μ among all the measures in C.

Proof. Let D := W (μ, C), let (ν_{n}) be a minimizing sequence, choose γ_{n} ∈ Opt(μ, ν_{n}) for any n inℕ and fix α_{n}^{m} ∈ ADM_{μ}(γ_{n}, γ_{m}). Since π_{#}^{1,2}α_{n}^{m} ∈ Adm(ν_{n}, ν_{m}) we have (π^{1}/2 + π^{2}/2)_{#}α_{n}^{m} ∈ C, therefore the Hilbertian identity

|\frac{b − c}{2}|^{2} = \frac{1}{2}|a − b|^{2} + \frac{1}{2}|a − c|^{2} − |a − \frac{b + c}{2}|^{2} ,

yields to

\begin{matrix}\frac{1}{4}W^{2}(ν_{n}, ν_{m}) ≤ & \frac{1}{4}∥x_{1} − x_{2}∥_{α_{n}^{m}}^{2} \\ = & \frac{1}{2}∥x_{0} − x_{1}∥_{α_{n}^{m}}^{2} + \frac{1}{2}∥x_{0} − x_{2}∥_{α_{n}^{m}}^{2} − ||x_{0} − \frac{x_{1} + x_{2}}{2} ||_{α_{n}^{m}}^{2} \\ ≤ & \frac{1}{2}W^{2}(μ, ν_{n}) + \frac{1}{2}W^{2}(μ, ν_{m}) − D^{2}. \end{matrix}

It follows that (v_{n}) is a Cauchy sequence, and given that C is W −closed, that it converges to a point v ∈ C which is clearly a (and actually the) point of minimal distance. □

Now we turn to the main result of this section.

Theorem 5.8. Let C ⊂ 𝒫_{2}(ℝ^{d}) be a W −closed strongly geodesically convex set. Then it is τ−closed.

Proof. Fix a measure μ not in C, let ν be the minimizing point found by the previous proposition and choose a plan γ ∈ exp_{μ}^{−1}(ν). We will show that C ⊂ ℋ_{γ;D^{2}}^{+}, where D := W (μ, C): since clearly μ is outside that halfspace, by proposition 5.4 and the arbitrariness of μ we conclude.

Choose any σ ∈ C, a plan σ ∈ exp_{μ}^{−1}(σ) and a 3-plan α ∈ ADM_{μ}(γ, σ). Since μ_{t} := (π^{0} + (1 − t)π^{1} + tπ^{2})_{#}α ∈ C, the minimality property of ν yields to

\begin{matrix}0 ≤ \frac{d^{+}}{dt} \frac{1}{2}W^{2}(μ_{t}, μ)|_{t=0} ≤ & \frac{d^{+}}{dt} \frac{1}{2}∥(1 − t)x_{1} + tx_{2}∥_{α}^{2}|_{t=0} \\ = & ⟨x_{2} − x_{1}, x_{1}⟩_{α} = ⟨x_{2}, x_{1}⟩_{α} − ∥γ∥^{2} ≤ ⟨γ, σ⟩ − D^{2}. \end{matrix}

By the arbitrariness of σ the thesis is achieved. □

3. Geodesically convex subsets of 𝒫_{2}^{r}(ℝ^{d})

In this section we drop strong assumption 5.2 on the W −closed set C, substituting it with the weaker one 5.1. Furthermore, we assume that every measure in C is regular, and we will prove that C is weakly closed. It is not clear to the author if the thesis remains true dropping the regularity assumption.

3. GEODESICALLY CONVEX SUBSETS OF 𝒫_{2}^{r}(ℝ^{d}) 77

This case is much more complex than the one analyzed in the previous section: in order to clarify the ides behind the proof, we recall here the main steps.

step 1 We prove that without loss of generality we can assume to deal with geodesically convex sets which are bounded: indeed recalling the definition of the topology τ, we have that a subset C of 𝒫_{2}(ℝ^{d}) is τ−closed if and only if so are its intersections with the closed ball and center in δ_{0}. Proving that those intersections are closed, geodesically convex and bounded we will get the claim.

step 2 This is the most important step of the proof: we will show that for each closed subset C of a complete geodesic metric space (E, d), every a < 1 and every point x ∉ C there exists a geodesic from C to x “which moves away from C with velocity bigger than a”, that is, there exists a geodesic parameterized with arc length t → x_{t} such that x_{0} ∈ C and x_{d(x_{0},x)} = x satisfying

\frac{d(x_{t}, C)}{t} ≥ a, ∀t ∈ (0, d(x_{0}, x)].

The interest of this claim comes from the lack of any kind of compactness hypothesis. Observe that if C is compact, the claim follows easily by picking any point y ∈ C of minimal distance from x and any geodesic from y to x. Such a geodesic would have the stated property even for a = 1.

step 3 We assume that every measure in our bounded, geodesically convex, closed set C is regular. Under this assumption we will show that the set C may be written as the intersection of a family of weakly closed halfspaces. More in detail: we will use the previous step to show that for each measure μ ∉ C and every a < 1 we can find a measure μ^{a} such that t → μ_{t}^{a} := (Id + t(T_{μ_{a}}^{μ} − Id))_{#}μ_{a} satisfies W (μ_{t}^{a}, C) ≥ at W (μ_{a}, μ). Then the geodesic convexity (this is the point in which we use this hypothesis) implies, with some calculation, that it holds

⟨T_{μ_{a}}^{μ} − Id, T_{μ_{a}}^{ν} − Id⟩ ≤ C_{a}, ∀ν ∈ C,

where C_{a} := \sqrt{1 − a^{2}}R W (μ_{a}, μ) and R is the diameter of C. Since C_{a} → 0 as a → 1, we can find some a such that the set C is contained in the “halfspace” given by

ℋ_{T_{μa}^{μ}−Id;C_{a}}^{−} := {ν : ⟨T_{μ_{a}}^{μ} − Id, T_{μ_{a}}^{ν} − Id⟩_{μ_{a}} ≤ C_{a}} ,

while the measure μ is not.

Showing that such a set is weakly closed we will get the conclusion.

Let us now enter into details.

Proposition 5.9 (Step 1). Let C be a geodesically convex subset of 𝒫_{2}(ℝ^{d}). Then letting B_{R} := {μ : ∥x∥_{μ} ≤ R}, we have that C ∩ B_{R} is geodesically convex for every R ∈ ℝ^{+}. Moreover C is τ−closed if and only if C ∩ B_{R} is τ−closed for every R ∈ ℝ^{+}, where B_{R} := {μ : ∥x∥_{μ} ≤ R}.

Proof. We prove at first that C ∩ B_{R} is geodesically convex. Let μ_{1}, μ_{2} ∈ C ∩ B_{R} and find an optimal plan γ such that μ_{t} := γ[t] ∈ C for every t ∈ [0, 1]. It holds

∥x∥_{μ_{t}}^{2} = ∥(1 − t)x_{1} + tx_{2}∥_{γ}^{2} ≤ (1 − t)∥x_{1}∥_{γ}^{2} + t∥x_{2}∥_{γ}^{2} = (1 − t)∥x_{1}∥_{μ_{1}}^{2} + t∥x∥_{μ_{2}}^{2} ≤ R^{2},

which is the first claim.

The other claim follows directly from the definition of the topology τ and from proposition 2.15. □

Proposition 5.10 (Step 2). Let (E, d) be a complete geodesic metric space, C ⊂ E a closed set, P ∈ E \ C, and 0 ≤ a < 1. Then there exists a point Q ∈ C such that

(5.1)  \frac{d(Q_{t}, C)}{t} ≥ a, ∀t ∈ (0, d(P, Q)],

where Q_{t} : [0, d(P, Q)] → E is any choice of a geodesic connecting Q to P (Q_{0} = Q, Q_{1} = P) parameterized by arc length.

Proof. Let we fix a notation: for any point R ∈ C let R be the set of geodesics connecting R to P parameterized by arc length, and let R_{t} ∈ R be a generic element of this set.

We will say that a point R ∈ C has the property G iff for every R_{t} ∈ R it holds

\frac{d(R_{t}, C)}{t} ≥ a, ∀t ∈ (0, d(P, R)].

Our aim is to prove that a point with the property G exists.

Start choosing any point R ∈ C and suppose that it doesn’t have the property G. Then there exists a point R′ ∈ C such that

(5.2)  d(R_{t}, R′) < at, for some t > 0 and some R_{t} ∈ R.

From this we get

d(R, R′) ≤ d(R, R_{t}) + d(R_{t}, R′) < t(a + 1),

and d(R′, P) ≤ d(R′, R_{t}) + d(R_{t}, P) < at + d(R, P) − t from which it follows

(5.3)  d(R, P) − d(R′, P) > t(1 − a).

Putting together the last two inequalities we get the key estimate

(5.4)  d(R, R′) < \frac{1 + a}{1 − a} (d(R, P) − d(R′, P)) .

This inequality is all we need to prove the thesis: we will proceed by transfinite induction by using its telescopic property.

Let Ω be the first uncountable ordinal. Define a function

\begin{aligned}Ω & → C \\ α & → R_{α}, \end{aligned}

beginning by choosing R_{0} ∈ C in any way. Then if α is the successor of some ordinal, we have two cases:

i) R_{α−1} has the property G, ii) R_{α−1} does not have the property G.

In the first case we put R_{α} := R_{α−1}, in the second one we choose R_{α} among those points R′ satisfying (5.2) with R = R_{α−1}. Finally, if α is a limit ordinal we let R_{α} be the limit of R_{α′} with α′ < α.

We have to prove this to be a good definition, we will do this by proving at the same tima that the following “extended” version of (5.4) holds:

(5.5)  d(R_{α}, R_{β}) < \frac{1 + a}{1 − a} (d(R_{α}, P) − d(R_{β}, P)) , ∀α ≤ β.

We prove this inequality by transfinite induction on β: it is true for 0, and it is easy to see that if it holds for β then it holds for β + 1. Indeed by construction and from the first part of the proof, R_{β+1} satisfies (5.4) with R = R_{β}, R′ = R_{β+1}, therefore combining (5.4) and (5.5) we get

d(R_{α}, R_{β+1}) ≤ d(R_{α}, R_{β}) + d(R_{β}, R_{β+1})

< \frac{1 + a}{1 − a} (d(R_{α}, P) − d(R_{β}, P) + d(R_{β}, P) − d(R_{β+1}, P))

= \frac{1 + a}{1 − a} (d(R_{α}, P) − d(R_{β+1}, P)) , ∀α ≤ β.

Given that the case α = β + 1 is obvious, we get the claim.

Now let β be a limit ordinal, observe that we can’t write inequality (5.5) for such a β, yet, since we have still to prove that R_{β} exists: we are going to prove at the same time that R_{β} is well defined and that for this point (5.5) holds. Since β < Ω there exists an increasing sequence (α_{n}) converging to β; for every α_{n} the inequality (5.5) holds, therefore we have

d(R_{α_{m}}, R_{α_{n}}) < \frac{1 + a}{1 − a} (d(R_{α_{m}}, P) − d(R_{α_{n}}, P)) , ∀m ≤ n.

Being the sequence d(R_{α_{n}}, P) non increasing (by equation (5.3)) and bounded from below, it is a Cauchy sequence and the previous inequality shows that the same is true for the sequence R_{α_{n}}, which therefore converges to some point we call R_{β}. Since the previous argument applies to every increasing sequence α_{n}, showing that the corrispond points R_{α_{n}} form a Cauchy sequence, we get that R_{β} is well defined (i.e. it does not depend on the particular sequence (α_{n}) chosen), that the function α → R_{α} is continuous (with respect to the order topology) and that (5.5) holds for any β < Ω.

Observe that from inequality (5.3) it follows that if R_{α+1} ≠ R_{α}, then d(R_{α+1}, P) is strictly less than d(R_{α}, P). We are almost done: since there is no strictly decreasing function from Ω to ℝ, we have that the map α → R_{α} has to be eventually constant, therefore for some α we have R_{α} = R_{α+1}, which means by construction that the point Q = R_{α} satisfies the thesis. □

Observe that we proved a statement stronger than the one climed at the beginning of the section: indeed we proved not only that there exists a geodesic from some point C to P with the stated property, but also that there exists a point Q such that every geodesic connecting it to P has the property.

Note that this proposition is a generalization of the Drop Theorem of Daneš valid in Banach spaces, see [21] for further reference.

After the work on thesis was completed, the author discovered that both the statement of the proposition and its proof are closely related to the Ekeland-Bishop-Phelps principle. Actually a shorter proof may be given with a direct application of the EBP principle: we present here one found by B.Kirchheim. Use EBP principle to find Q ∈ C which is a minimizer of

x → f(x) := d(x, P) + \frac{1 − a}{1 + a}d(x, Q).

Then such a Q has the claimed property. Indeed, if this is not the case, there exists R ∈ C and 0 ≤ t ≤ d(P, Q) such that d(R, Q_{t}) < at. For such R we have the following bounds

\begin{aligned}d(R, P) & ≤ d(R, Q_{t}) + d(Q_{t}, P) < at + d(P, Q) − t, \\ d(R, Q) & ≤ d(R, Q_{t}) + d(Q_{t}, Q) < at + t. \end{aligned}

Therefore it holds

f(Q) = d(P, Q) = d(P, Q) − t(1 − a) + \frac{1 − a}{1 + a}t(a + 1) > d(R, P) + \frac{1 − a}{1 + a}d(R, Q) = f(R),

which contradicts the minimality of Q.

Hilbert case 3. Let C be a strongly closed, convex subset of an Hilbert space H. Then C is weakly closed.

Proof. Since C is convex and weakly closed if and only if so are the sets C ∩ B_{R}, we may assume without loss of generality that C is bounded. Let R be its diameter. Choose any point y ∉ C: the claim will be achieved if we show that there exist a point x and a constant c ∈ ℝ such that

⟨z − x, y − x⟩ ≤ c < |y − x|^{2}, ∀z ∈ C.

Indeed in this case the set C would be included in the halfspace {z : ⟨z − x, y − x⟩ ≤ c} which is weakly closed and does not contain y. By the arbitrarieness of y this would give the thesis.

Let us prove our claim. Fix a < 1 and apply proposition 5.10 with P = y to find a point x^{a} ∈ C satisfying

(5.6)  inf_{z∈C} |(1 − t)x^{a} + ty − z| ≥ at|y − x^{a}|.

Now fix z ∈ C and let v^{a} := y − x^{a}, w := z − x^{a}. Observe that for small t > 0 it holds

x^{a} + t\sqrt{1 − a^{2}} \frac{|v^{a}|}{|w|}w ∈ C,

therefore we know from (5.6) that

|tv^{a} − t\sqrt{1 − a^{2}} \frac{|v^{a}|}{|w|}w|^{2} ≥ a^{2}t^{2}|v^{a}|^{2}.

Some manipulations show that the last equation implies

⟨v^{a}, w⟩ ≤ |v^{a}||w|\sqrt{1 − a^{2}} ≤ R|v^{a}|\sqrt{1 − a^{2}}.

By choosing a near to 1 and observing that |v^{a}| ≤ d(y, C)+R we get that the last term of the previous inequality is close to 0. Therefore it is smaller than d^{2}(y, C) which in turn is smaller than |y − x^{a}|^{2} and the claim is achieved. □

Proposition 5.11 (Step 3). Let C be a strongly closed, geodesically closed subset of 𝒫_{2}^{r}(ℝ^{d}). Then C is τ−closed.

Proof. Applying proposition 5.9 we may assume without loss of generality that C is bounded. Let R be its diameter. Choose any measure ν ∉ C: the claim will be achieved if we show that there exists a measure μ ∈ C and a constant c ∈ ℝ such that

⟨T_{μ}^{ν} − Id, T_{μ}^{σ} − Id⟩ ≤ c < ∥T_{μ}^{ν} − Id∥_{μ}^{2} = W^{2}(μ, ν), ∀σ ∈ C.

Indeed in this case the set C would be included in the halfspace ℋ_{T_{μ}^{ν}−Id;c}^{−} which is weakly closed by proposition 5.4 and does not contain ν. By the arbitrariness of ν we can conclude.

Let us prove our claim. Fix a < 1 and apply proposition 5.10 with P = ν to find a measure μ^{a} satisfying

W (μ_{t}^{a}, C) ≥ atW (ν, μ_{t}^{a}),

where μ_{t}^{a} := (Id+t(T_{μ^{a}}^{ν} −Id))_{#}μ^{a}. Now fix σ ∈ C and define v^{a} = T_{μ}^{ν} −Id, w = T_{μ}^{σ}−Id. Observe that for small t > 0 it holds

σ_{t} := (Id + t\sqrt{1 − a^{2}} \frac{∥v^{a}∥_{μ^{a}}}{∥w∥_{μ^{a}}} w)_{#} μ^{a} ∈ C,

therefore we know that

W^{2}(μ_{t}^{a}, σ_{t}) ≥ a^{2}t^{2}∥v^{a}∥_{μ^{a}}^{2}.

Recalling that for every couple of tangent plans γ_{1}, γ_{2} (see (4.18)) it holds

lim_{t→0^{+}} \frac{W (exp(t · γ_{1}), exp(t · γ_{2}))}{t} = W_{μ}(γ_{1}, γ_{2}),

and that v^{a}, w ∈ Tan_{μ^{a}}(𝒫_{2}(ℝ^{d})) we obtain

||v^{a} − \sqrt{1 − a^{2}} \frac{∥v^{a}∥_{μ^{a}}}{∥w∥_{μ^{a}}} w||_{μ^{a}}^{2} ≥ a^{2}∥v^{a}∥_{μ^{a}}^{2}.

Some algebraic manipulations show that the previous inequality implies

⟨v^{a}, \sqrt{1 − a^{2}} \frac{∥v^{a}∥_{μ^{a}}}{∥w∥_{μ^{a}}} w⟩_{μ^{a}} ≤ ∥v^{a}∥_{μ^{a}}∥w∥_{μ^{a}} \sqrt{1 − a^{2}} ≤ R∥v^{a}∥_{μ^{a}} \sqrt{1 − a^{2}}.

By choosing a near to 1 and observing that ∥v^{a}∥_{μ^{a}} ≤ W (ν, C) + R we get that the last term of the previous inequality is close to 0. Therefore it is smaller than d^{2}(ν, C), which in turn is smaller then W^{2}(ν, μ^{a}) and the claim is achieved. □

4. Subdifferentials and differentiability properties of W^{2}

We conclude this chapter with a short discussion on the right definition of subdifferential of geodesically convex functional (with the hope to clarify the one introduced in [11] in light of the notation introduced here) and to complete the analysis of the differentiability properties of the function W^{2}(·, σ). Although the results proven here are not needed in the sequel, they are interesting from a theoretical point of view, and provide an application of the theory developed up to here.

Definition 5.12 (geodesically convex functionals). A functional F : 𝒫_{2}(ℝ^{d}) → ℝ∪ {+∞} is geodesically convex if for every constant speed geodesic t → μ_{t} parameterised in some interval, the function

t → F (μ_{t}),

is convex.

Obviously the definition would be the same if we let the geodesics be parameterised in [0, 1]. Moreover, it is a simple consequence of theorem 2.8 that F is geodesically convex if and only if for any couple of measures μ, ν there exists a constant speed geodesic t → μ_{t} joining μ to ν along which F is convex.

In the following we will always deal with proper functionals, i.e. functionals which are not constantly equal to +∞.

A simple consequence of the theorem proved in the previous section is the following one.

Theorem 5.13. Let F be a geodesically convex functional, lower semicontinuous w.r.t. W. Assume also that F satisfies one of the two conditions below:

• F (μ_{t}) is convex along curves μ_{t} = [γ](t) even for non optimal γ;

• F (μ) = +∞ for any measure μ ∉ 𝒫_{2}^{r}(ℝ^{d}). Then F is lower semicontinuous w.r.t. the weak topology τ.

Proof. In either case the sublevels of F are weakly closed because of what we proved in the last two sections. □

We will denote by D(F) the domain of F, that is the set of those measure μ ∈ 𝒫_{2}(ℝ^{d}) such that F (μ) < +∞.

Definition 5.14 (Subdifferential, superdifferential and differential). Let F : 𝒫_{2}(ℝ^{d}) → (−∞, +∞] be a functional and let μ ∈ D(F). We say that a plan γ ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) belongs to the subdifferential ∂^{−}F (μ) of F at μ if for every measure ν and every η ∈ exp_{μ}^{−1}(ν) it holds

(5.7)  F (ν) ≥ F (μ) − ⟨ − 1 · γ, η⟩ + o(W (ν, μ)).

Analogously, we say that γ ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) belongs to the superdifferential ∂^{+}F (μ) of F at μ if for every measure ν and every η ∈ exp_{μ}^{−1}(ν) it holds

(5.8)  F (ν) ≤ F (μ) + ⟨γ, η⟩ + o(W (ν, μ)).

4. SUBDIFFERENTIALS AND DIFFERENTIABILITY PROPERTIES OF W^{2} 83

A plan γ ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) is said to be the differential ∂F (μ) of F at μ if it is both a subdifferential and a superdifferential (it is easy to see that a differential, whenever it exists, is unique).

We will denote by D(∂^{±}F) the domain superdifferential and subdifferential respectively, i.e. the set of those measures μ such that the set ∂^{±}F (μ) is not empty.

Observe that for regular measures μ the previous definitions reduce to:

\begin{aligned}v ∈ ∂^{−}F (μ) & ⇔ F (ν) ≥ F (μ) + ⟨v, T_{μ}^{ν} − Id⟩_{μ} + o(W (ν, μ)), ∀ν ∈ 𝒫_{2}(ℝ^{d}) \\ v ∈ ∂^{+}F (μ) & ⇔ F (ν) ≤ F (μ) + ⟨v, T_{μ}^{ν} − Id⟩_{μ} + o(W (ν, μ)), ∀ν ∈ 𝒫_{2}(ℝ^{d}). \end{aligned}

A natural question arises: why the term −⟨ − 1 · γ, η⟩ instead of ⟨γ, η⟩ in the definition of sub-differential? The answer is that the definition we have chosen is the right one to ensure the strong-weak closure of subdifferentials of geodesically convex functions: this will become clear with the proof of theorem 5.18 and example 5.19. Here we just observe that by the analogy with the Euclidean case a priori one assumes that for regular measures the linear term has to be similar to ⟨v, T_{μ}^{ν} − Id⟩ (which is actually the case) while there is no reason a priori to guess that when dealing with non-regular measures μ, the correct generalization is ⟨γ, η⟩ instead of some other member of the set ⟨x_{1}, x_{2}⟩_{α}, when α varies in the union of the sets ADM_{μ}(γ, η) as η varies in exp_{μ}^{−1}(ν). If exp_{μ}^{−1}(ν) contains only one element η, one may just hope for the right term to be either

max_{α}⟨x_{1}, x_{2}⟩_{α}, α ∈ ADM_{μ}(γ, η) or,
min_{α}⟨x_{1}, x_{2}⟩_{α}, α ∈ ADM_{μ}(γ, η),

which lead respectively to (see also remark 4.28)

\begin{matrix}⟨ & γ, η⟩, \\ −⟨ & − 1 · γ, η⟩. \end{matrix}

Choosing a priori one definition instead of the other one is just a matter of taste, but, as we said before, the second one leads to the properties of the subdifferential we would like to have.

Observe that by virtue of our definition in order to prove that equation (5.7) is satisfied one has just to prove that equation

F (ν) ≥ F (μ) + ⟨x_{1}, x_{2}⟩_{α} + o(W (ν, μ)),

holds for some α ∈ ADM_{μ}(γ, η) with η ∈ exp_{μ}^{−1}(ν).

In case we are dealing with geodesically convex lower semicontinuous functionals, the definition of subdifferential simplies a bit, as showed by the following proposition.

Proposition 5.15. Let F be a g.c. and l.s.c. functional and let μ ∈ D(F) be a measure in its domain. Then γ ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) belongs to ∂^{−}F (μ) if and only if for every ν ∈ 𝒫_{2}(ℝ^{d}) there exists η ∈ exp_{μ}^{−1}(ν) such that

(5.9)  F (ν) ≥ F (μ) − ⟨ − 1 · γ, η⟩.

Proof. We have to prove that we can drop the term o(W (μ, ν)) and that it is sufficient the existence of one element in exp_{μ}^{−1}(ν) satisfying (5.9), instead than asking the inequality to hold for any plan in exp_{μ}^{−1}(ν).

Let us first check the second property: take any η ∈ exp_{μ}^{−1}(ν) and consider the curve t → ν_{t} := exp(t · η) for 0 ≤ t ≤ 1. We know that for t < 1 there exists only one element η_{t} in exp_{μ}^{−1}(ν_{t}), namely (π^{0}, π^{0} + tπ^{1})_{#}η, therefore it has to hold

(1 − t)F (μ) + tF (ν) ≥ F (ν_{t}) ≥ F (μ) − ⟨ − 1 · γ, η_{t}⟩ + o(W (ν_{t}, μ)),

which leads to the claim by letting t go to 1.

For the other part of the proof, argue by contradiction, assuming that

F (ν) ≤ F (μ) − ⟨ − 1 · γ, η⟩ − ε,

for some ν ∈ 𝒫_{2}(ℝ^{d}), η ∈ exp_{μ}^{−1}(ν) and ε > 0. Then, defining ν_{t} := exp_{μ}(t · η) we get that

F (ν_{t}) ≤ (1 − t)F (μ) + tF (ν) ≤ F (μ) − ⟨ − 1 · γ, t · η⟩ − tε,

which contradicts equation (5.7) by letting t go to 0. □

The definition of geodesically convex sets given before for subsets of 𝒫_{2}(ℝ^{d}) has an analogous for subsets of Tan_{μ}(𝒫_{2}(ℝ^{d})): a set K ⊂ Tan_{μ}(𝒫_{2}(ℝ^{d})) is geodesically convex if for every couple of plans γ_{1}, γ_{2} ∈ K there exists a geodesic connecting them entirely contained in K. In other words, for any γ_{1}, γ_{2} ∈ K there has to exist an α ∈ OPT_{μ}(γ_{1}, γ_{2}) such that α[t] ∈ K for any 0 ≤ t ≤ 1.

Proposition 5.16. Let F be a functional on 𝒫_{2}(ℝ^{d}). Then for every μ ∈ 𝒫_{2}(ℝ^{d}) the set ∂^{−}F (μ) is τ−sequentially closed and geodesically convex.

Proof. We can assume that ∂^{−}F (μ) is not empty. To get the convexity apply proposition 4.27 to get that

(5.10)  ⟨((1 − t) · γ_{1}) ⊕ (t · γ_{2}), η⟩ ≤ ⟨(1 − t) · γ_{1}, η⟩ + ⟨t · γ_{2}, η⟩.

To prove the closure take any sequence γ_{n} ∈ ∂^{−}F (μ) τ−converging to some γ, fix a plan η ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) and consider any sequence (α_{n}) ⊂ OPT_{μ}(−1 · γ_{n}, η). Then, possibly extracting a subsequence, we can assume that (α_{n}) τ−converges to some α ∈ ADM_{μ}(−1 · γ, η). Then it holds

lim_{n→∞} ⟨ − 1 · γ_{n}, η⟩ = lim_{n→∞} ⟨x_{1}, x_{2}⟩_{α_{n}} = ⟨x_{1}, x_{2}⟩_{α} ≤ ⟨ − 1 · γ, η⟩.

□

Observe that we proved a stronger kind of convexity for the set ∂^{−}F (μ): indeed equation (5.10) shows that for any α ∈ ADM_{μ}(γ_{1}, γ_{2}) (and not just for those in OPT_{μ}(γ_{1}, γ_{2})) the whole segment α[t] belongs to ∂^{−}F (μ). The weak closure of subdifferential will be more deeply analyzed in theorem 5.18 below.

Corollary 5.17. Let F be a functional and μ ∈ D(∂^{−}F). Then there exists a unique element in ∂^{−}F (μ) of minimal norm.

Proof. The existence is a simple consequence of the τ−compacteness of bounded sequences in Tan_{μ}(𝒫_{2}(ℝ^{d})) and of the previous proposition. The uniqueness follows from the strict convexity of the norm on Tan_{μ}(𝒫_{2}(ℝ^{d})): suppose that there exist two plans γ_{1}, γ_{2} of minimal norm and take any α ∈ OPT_{μ}(γ_{1}, γ_{2}). Then the previous proposition ensures that for any 0 < t < 1 the plan α[t] belongs to ∂^{−}F (μ). For such plans it holds

∥α[t]∥^{2} = ∥(1 − t)x_{1} + tx_{2}∥_{α}^{2} < (1 − t)∥x_{1}∥_{α}^{2} + t∥x_{2}∥_{α}^{2} = (1 − t)∥γ_{1}∥^{2} + t∥γ_{2}∥^{2},

contradicting the minimality. □

The following theorem shows that even our definition of subdifferential, reproduces the classical strong-weak closure property. A counterexample follows, showing that the other alternative doesn’t work as well.

Theorem 5.18 (Strong-weak closure of subdifferential). Let F be a g.c. and l.s.c. functional, μ_{n}, μ ∈ 𝒫_{2}(ℝ^{d}), n ∈ ℕ, be given measures such that W (μ_{n}, μ) → 0 as n → ∞ and let γ_{n} ∈ ∂^{−}F (μ_{n}) be τ converging to some γ. Then γ ∈ ∂^{−}F (μ).

Proof. Obviously π_{#}^{1}γ = μ. Fix ν ∈ 𝒫_{2}(ℝ^{d}) and observe that the following inequality holds for every n:

F (ν) ≥ F (μ_{n}) − ⟨ − 1 · γ_{n}, η_{n}⟩,

where η_{n} is a certain plan in exp_{μ_{n}}^{−1}(ν). Since F is lower semicontinuous, to get the thesis we have just to prove that

\overline{lim}_{n} ⟨ − 1 · γ_{n}, η_{n}⟩ ≤ ⟨ − 1 · γ, η⟩,

where η is some element of exp_{μ}^{−1}(ν). Possibly extracting a subsequence we can assume that (η_{n}) W −converge w.r.t. W to some η such that π_{#}^{1}η = μ, (π^{1} + π^{2})_{#}η = ν, moreover since the plans (π^{1}, π^{1} + π^{2})_{#}η_{n} are optimal, the same is true for η, which therefore belongs to exp_{μ}^{−1}(ν). Now choose α_{n} ∈ OPT_{μ}(−1 · γ_{n}, η_{n}) and take a subsequence (not relabeled) which τ converges to some α ∈ ADM_{μ}(−1 · γ, η). Since π_{#}^{1,3}α_{n} = η_{n} converge w.r.t. W to π_{#}^{1,3}α = η we can apply corollary 2.29 to get

lim_{n→∞} ⟨ − 1 · γ_{n}, η_{n}⟩ = lim_{n→∞} ⟨x_{1}, x_{2}⟩_{α_{n}} = ⟨x_{1}, x_{2}⟩_{α} ≤ ⟨ − 1 · γ, η⟩.

□

The proof of the previous theorem suggests why the same technique doesn’t work with the other possible definition of subdifferential: with the same notation used in the proof, if one tries to prove that \underline{lim} ⟨γ_{n}, η_{n}⟩ ≥ ⟨γ, η⟩ by taking α_{n} ∈ OPT_{μ}(γ_{n}, η_{n}) and passing to the limit, one still has that ⟨x_{1}, x_{2}⟩_{α_{n}} = ⟨x_{1}, x_{2}⟩_{α}, but nothing ensures that the limit 3-plan α is optimal for the couple γ, η. The following example shows that there is actually no hope to find a proof of the previous theorem with the other definition of subdifferential.

Example 5.19. For t ≥ 0 let μ_{t} := (δ_{t} + δ_{−t})/2 ∈ 𝒫_{2}(ℝ) and define the functional F as

F (μ) := \{ \begin{matrix}−t, & if μ = μ_{t} for some 0 ≤ t ≤ 1, \\ +∞, & elsewhere. \end{matrix}

It is not difficult to see that F is a g.c. and l.s.c. functional and that for 0 < t ≤ 1 γ_{t} := (δ_{(t,−1)} + δ_{(−t,1)})/2 belongs to ∂^{−}F (μ_{t}) (actually it is the unique element): indeed for fixed t, T > 0 let η_{t}^{T} := (δ_{(t,T −t)} +δ_{(−t,−T +t)})/2 be the unique element in exp_{μ_{t}}^{−1}(μ_{T}). Then for every 0 < t, T ≤ 1 it holds ⟨γ_{t}, η_{t}^{T} ⟩ = t − T and therefore the inequality

F (μ_{T}) ≥ F (μ_{t}) + ⟨γ_{t}, η_{t}^{T} ⟩

holds (and it is an equality). But letting t go to 0, clearly γ_{t} converges to γ := (δ_{(0,1)} + δ_{(0,−1)})/2 and we have that ⟨γ, η_{0}^{T} ⟩ = T, which is strictly bigger than the limit of ⟨γ_{t}, η_{t}^{T} ⟩, and the previous equation does not hold:

F (μ_{T}) = −T ≱ T = F (μ_{0}) + ⟨γ_{0}, η_{0}^{T} ⟩.

Now we turn to the study of differential properties of W^{2}. Observe that by inequality (2.16) we know that W^{2}(·, σ) satisfies a concavity inequality (it is -1 geodesically concave, in a sense which can be specified), therefore we should expect to find superdifferentiability properties rather than subdifferentibility ones. The following theorem shows that actually the superdifferential is always not empty.

Theorem 5.20. Let μ, σ ∈ 𝒫_{2}(ℝ^{d}) be two measures and fix γ ∈ exp_{μ}^{−1}(σ). Then −2 · γ belongs to the superdifferential of the functional ν → W^{2}(ν, σ) at μ.

Proof. Choose ν ∈ 𝒫_{2}(ℝ^{d}) and η ∈ exp_{μ}^{−1}(ν) and α ∈ OPT_{μ}(η, γ). Then (π^{0} + π^{1}, π^{0} + π^{2})_{#}α ∈ Adm(ν, σ) and therefore it holds

\begin{matrix}W^{2}(ν, σ) ≤ ∥x_{1} − x_{2}∥_{α}^{2} = & ∥x_{1}∥_{α}^{2} + ∥x_{2}∥_{α}^{2} − 2⟨x_{1}, x_{2}⟩_{α} \\ = & W^{2}(μ, ν) + W^{2}(μ, σ) − ⟨2 · γ, η⟩, \end{matrix}

from which the thesis follows. □

Once it is proved that the superdifferential is always not empty the natural question that arises is: what is the element of minimal norm? The non trivial answer is given in the following theorem.

Theorem 5.21. Let μ, σ ∈ 𝒫_{2}(ℝ^{d}) be two measures. Then the element of minimal norm of the superdifferential of W^{2}(·, σ) at μ is −2ℬ(γ_{0}), where γ_{0} is the unique minimiser of ∥ℬ(γ)∥_{μ} among all γ in exp_{μ}^{−1}(σ).

Proof. Let us first prove that there is only one minimiser of ∥ℬ(γ)∥_{μ} in exp_{μ}^{−1}(σ). The existence follows by τ−compacteness of the set and lower semicontinuity of the norm of the barycentric projection. For uniqueness, observe that exp_{μ}^{−1}(σ) is stable under (usual) convex combination of plans, and that t → ∥ℬ((1 − t)γ_{1} + tγ_{2})∥_{μ} = ∥(1 − t)ℬ(γ_{1}) + tℬ(γ_{2})∥_{μ} is strictly convex.

To prove the thesis we will first show that −2ℬ(γ) is a superdifferential for every γ ∈ exp_{μ}^{−1}(σ), and then we prove that for every superdifferential γ it holds P_{μ}(ℬ(γ)) = −2ℬ(η) for some η ∈ exp_{μ}^{−1}(σ). Given that it holds ∥P_{μ}(ℬ(γ))∥_{μ} ≤ ∥ℬ(γ)∥_{μ} ≤ ∥γ∥ this will prove the thesis.

To prove the first claim choose a plan γ ∈ exp_{μ}^{−1}(σ), a measure ν, a plan η ∈ exp_{μ}^{−1}(ν) and define the 3-plan α to be the plan whose disintegration w.r.t. μ satisfies α_{x_{0}} = γ_{x_{0}} × η_{x_{0}}. Observe that α satisfies (π^{0} + π^{1}, π^{0} + π^{2})_{#}α ∈ Adm(σ, ν) and ⟨x_{1}, x_{2}⟩_{α} = ⟨ℑ(ℬ(γ)), η⟩, therefore it holds

W^{2}(ν, σ) ≤ ∥x_{1} − x_{2}∥_{α}^{2} = W^{2}(μ, σ) + W^{2}(μ, ν) − 2⟨ℑ(ℬ(γ)), η⟩,

and the claim is proved.

To prove the second claim let us introduce the set

K := {−2ℬ(η) : η ∈ exp_{μ}^{−1}(σ)} ,

and observe that K is closed, convex and included in Tan_{μ}(𝒫_{2}(ℝ^{d})) (because of theorem 4.15). Therefore, in order to prove that P_{μ}(ℬ(γ)) ∈ K for every superdifferential γ in μ, by the Hanh-Banach theorem applied to in Tan_{μ}(𝒫_{2}(ℝ^{d})) it is sufficient to show that

(5.11)  ⟨ℬ(γ), v⟩_{μ} ≥ min_{f∈K} ⟨f, v⟩_{μ}, ∀v ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})).

By density it is sufficient to show that the previous inequality holds for every v such that (Id, Id + T v)_{#}μ is the unique optimal plan between its marginals for a certain T > 0. Let us assume this is the case, define μ_{t} := (Id + tv)_{#}μ and use proposition 4.10 to compute

\frac{d}{dt}W^{2}(μ_{t}, σ)|_{t=0} = − 2 sup_{η∈exp^{−} 1_{μ}(σ)} ⟨ℑ(v), η⟩
= − 2 sup_{η∈exp^{−} 1_{μ}(σ)} ⟨v, ℬ(η)⟩_{μ} = inf_{f∈K} ⟨v, f⟩_{μ}.

(5.12)

The same derivative can be evaluated using the fact that γ is a superdifferential, which gives the inequality (recall that (Id, Id + tv)_{#}μ is the unique optimal plan between μ and μ_{t} for t ≤ T):

W^{2}(μ_{t}, σ) ≤ W^{2}(μ, σ) − ⟨ − 1 · γ, ℑ(tv)⟩ + o(t) = W^{2}(μ, σ) + t⟨ℬ(γ), v⟩_{μ} + o(t).

This leads to

(5.13)  \frac{d}{dt}W^{2}(μ_{t}, σ)|_{t=0} ≤ ⟨ℬ(γ), v⟩_{μ}.

Equations (5.12) and (5.13) give (5.11), and therefore the thesis. □

The last two theorems allow the characterization of differentiability points of W^{2}(·, σ).

Corollary 5.22. The functional W^{2}(·, σ) is differentiable in μ if and only if the set exp_{μ}^{−1}(σ) contains only one element, and this element is induced by a map.

Proof. It is clear that if the superdifferential contains more than one element, then the point can’t be a point of differentiability. Therefore, given that by the previous theorems we know that both −2·γ and −2ℬ(γ) belong to the superdifferential at μ for every γ ∈ exp_{μ}^{−1}(σ), the points which do not satisfy the assumption of the statement are not point of differentiability.

To conclude, fix a point μ such that exp_{μ}^{−1}(σ) contains only an element γ induced by a map, choose a sequence of points ν_{n} ∈ 𝒫_{2}(ℝ^{d}), plans η_{n} ∈ exp_{μ}^{−1}(ν_{n}) and 3-plans α_{n} such that (π^{0}, π^{1} − π^{0})_{#}α_{n} = η_{n} and (π^{1}, π^{2} − π^{1})_{#}α_{n} ∈ exp_{ν_{n}}^{−1}(σ). We have

\begin{matrix}W^{2}(ν_{n}, σ) − W^{2}(μ, σ) ≥ & ∥x_{2} − x_{1}∥_{α_{n}}^{2} − ∥x_{2} − x_{0}∥_{α_{n}}^{2} \\ = & ∥x_{1} − x_{0}∥_{α_{n}}^{2} − 2⟨x_{1} − x_{0}, x_{2} − x_{0}⟩_{α_{n}} \\ ≥ & W^{2}(μ, ν_{n}) − 2⟨η_{n}, γ_{n}⟩, \end{matrix}

where γ_{n} := (π^{0}, π^{2} − π^{0})_{#}α_{n}. Given that (π^{0}, π^{0} + π^{1})_{#}γ_{n} ∈ Adm(μ, σ) and that ∥γ_{n}∥ ≤ ∥x_{1} − x_{0}∥_{α_{n}} + ∥x_{2} − x_{1}∥_{α_{n}} = W (μ, ν_{n}) + W (ν_{n}, σ) → W (μ, σ) we have that γ_{n} converges to γ w.r.t. W (because of the uniqueness of γ). By proposition 4.7 and the fact that γ is induced by a map, this implies that γ_{n} converges to γ w.r.t. W_{μ}, therefore the previous estimate gives

W^{2}(ν_{n}, σ) − W^{2}(μ, σ) ≥ W^{2}(μ, ν_{n}) − 2⟨η_{n}, γ⟩ + W (μ, ν_{n})o(1)

and the proof is achieved. □

CHAPTER 6 An approach to second order analysis

In the previous chapters we developed a first order theory for the space (𝒫_{2}(ℝ^{d}), W), showing that it admits a tangent space endowed with an inner product which makes 𝒫_{2}(ℝ^{d}) formally very similar to a Riemannian manifold. In order to go deeper in the study of this space some natural questions arise: is there a Levi-Civita connection for this Riemannian structure? Is there a parallel transport for a tangent vector along a curve? The aim of this chapter is to provide a first step in this direction and to show that under certain regularity assumptions both questions admit an affirmative answer. Part of the calculation made in this Chapter (more explicitely, part of those of section 5) already appeared in [43], where the equation of the covariant derivative and the expression for the curvature operator were computed at a formal level. The results proven here can be found in [10].

1. The case of a manifold embedded in ℝ^{d}

In this section we recall the Riemannian concepts we are going to study in the Wasserstein setting. An important ingredient is a proof of existence of the parallel transport that, as we will see later on, can be imitated to prove existence of the parallel transport along a dense class of curves in 𝒫_{2}(ℝ^{d}). For a deeper analysis of all the topics discussed in this section, apart the proof of existence of parallel transport we present, see [33].

Throughout this section M will be a C^{∞} manifold embedded in ℝ^{d} endowed with the induced Riemannian structure, γ(t) : [0, 1] → M a fixed C^{∞} curve and v(t) = γ̇(t) ∈ T_{γ(t)}M, t ∈ [0, 1], the velocity vector of γ(t). We will think to the tangent space V_{t} := T_{γ(t)}M at the point γ(t) as a linear subspace of ℝ^{d} (i.e. we translate it to let the origin be included) and we let P_{t} : ℝ^{d} → V_{t} be the orthogonal projection of ℝ^{d} onto V_{t}.

Recall that a connection on a M is a map (v, u) → D_{v}u on the set of smooth vector fields into itself, additive on each component and satisfying:

(6.1a)  D_{fv}u = fD_{v}u,
(6.1b)  D_{v}(fu) = fD_{v}u + ⟨∇f, v⟩u,

for any f ∈ C^{∞}(M).

Equation (6.1a) implies that for a given u, the value of D_{v}u at a point p depends only on the value of v at p, for this reason we will sometime write D_{v(p)}u for D_{v}u(p). Equation (6.1b) gives that for a given v, D_{v}u(p) depends only on the value of u along an integral curve γ(t) of v starting at p, i.e. a curve γ on a certain right-neighborhood of 0 such that γ(0) = p, γ′(t) = v(γ(t)).

In general there is more than one connection, however if the manifold is Riemannian, as in our case, only one is compatible with the metric and torsion free, i.e. only one satisfies the following two conditions:

(6.2a) \frac{d}{dt}(⟨u^{1}(γ(t)), u^{2}(γ(t))⟩) = ⟨∇_{v(t)}u^{1}(γ(t)), u^{2}(γ(t))⟩ + ⟨u^{1}(γ(t)), ∇_{v(t)}u^{2}(γ(t))⟩,
(6.2b)  ∇_{u^{1}}u^{2} − ∇_{u^{2}}u^{1} = [u^{1}, u^{2}].

where u^{1}, u^{2} are tangent vector fields, and in (6.2a) u^{1}(t), u^{2}(t) are their value in γ(t), respectively. The fact that there is at most one connection ∇_{v}(u) for which the previous equations are satisfied is a consequence of the Koszul formula:

2⟨∇_{v}(u), w⟩ = v(⟨u, w⟩) + u(⟨v, w⟩) − w(⟨u, v⟩) + ⟨[u, v], w⟩ − ⟨[u, w], v⟩ − ⟨[v, w], u⟩,

valid for any vector fields u, v, w defined on the whole M, and any connection ∇_{v}(u) satisfying equations (6.2) (in this formula we identified a tangent vector field with the derivation it induces). Given that the formula expresses the covariant derivative in terms of the Riemannian metric only, the uniqueness follows.

The unique connection satisfying (6.2) is called Levi-Civita connection. In our setting the Levi-Civita connection along a vector v ∈ T_{p}M tangent to M at the point p is given by the following formula:

(6.3)  ∇_{v}u(p) := P_{p} (\frac{d}{dt}u(γ(t))) ,

where u is a tangent vector field, γ is a smooth curve such that γ(0) = p, γ̇(0) = v and P_{p} is the orthogonal projection onto T_{p}M.

To prove that the connection we defined is the Levi-Civita one, observe that it holds

\frac{d}{dt}⟨u_{t}^{1}, u_{t}^{2}⟩ = ⟨\frac{d}{dt}u_{t}^{1}, u_{t}^{2}⟩ + ⟨u_{t}^{1}, \frac{d}{dt}u_{t}^{2}⟩ = ⟨∇_{v(t)}u^{1}(t), u^{2}(t)⟩ + ⟨u^{1}(t), ∇_{v(t)}u^{2}(t)⟩,

for any u_{t}^{1}, u_{t}^{2} tangent in γ(t), and

\begin{matrix}[u^{1}, u^{2}]F = & u^{1}(u^{2}(F)) − u^{2}(u^{1}(F)) = u^{1}(⟨∇F, u^{2}⟩) − u^{2}(⟨∇F, u^{1}⟩) \\ = & ⟨∇^{2}F · u^{1}, u^{2}⟩ + ⟨∇F, ∇_{u^{1}}u^{2}⟩ − ⟨∇^{2}F · u^{2}, u^{1}⟩ − ⟨∇F, ∇_{u^{2}}u^{1}⟩ \\ = & (∇_{u^{1}}u^{2} − ∇_{u^{2}}u^{1})F, \end{matrix}

for any two vector fields u^{1}, u^{2} and any smooth functional F on M (in the calculations we did, we implicitly assumed F to be extended to a neighborhood of M in such a way to leave ∇F (p) ∈ T_{p}M).

The vector field t → u_{t} is the parallel transport of the vector u_{0} along γ if and only if

(6.4)  P_{t} (\frac{d}{dt}u_{t}) = 0.

Observe that it is easy to prove the uniqueness of the solution of this equation: indeed by linearity it is sufficient to show that the norm is preserved in time, and this follows

1. THE CASE OF A MANIFOLD EMBEDDED IN ℝ^{d} 91

by:

\frac{d}{dt}∥u_{t}∥^{2} = 2⟨\frac{d}{dt}u_{t}, u_{t}⟩ = 2⟨P_{t}(\frac{d}{dt}u_{t}), u_{t}⟩ = 0.

Therefore the problem is to show the existence of a solution of (6.4) for a given initial datum u_{0}. This is usually done by using coordinates and solving an appropriate system of differential equations, however this technique cannot be applied to the space 𝒫_{2}(ℝ^{d}) (we have neither Cristoffel symbols, nor coordinates). Here we are going to show how the parallel transport can be constructed using tools which have a Wasserstein analogous.

One concept useful to understand the process we will use is the following:

Definition 6.1 (Angle between subspaces). Let V_{0}, V_{1} ⊂ ℝ^{d} be two given subspaces, and let P_{i}, i = 0, 1, be the orthogonal projection of ℝ^{n} onto V_{i} and P_{i}^{⊥} := Id − P_{i}. Then the angle θ(V_{0}, V_{1}) ∈ [0, π/2] is defined by the following formula:

sin θ(V_{0}, V_{1}) = ∥P_{1}^{⊥}|_{V_{0}}∥_{op} = sup_{{v_{0}∈V_{0} \atop |v_{0}|=1}} |P_{1}^{⊥}(v_{0})|.

It is not difficult to see that, letting V_{i}^{⊥}, i = 0, 1, the orthogonal complement of V_{i}, it holds

sin θ(V_{0}, V_{1}) = sup_{{v_{0}∈V_{0} \atop |v_{0}|=1}} |v_{0} − P_{1}(v_{0})|

= sup_{{v_{0}∈V_{0}, |v_{0}|=1 \atop v^{⊥}_{1}∈V^{⊥}_{1}, |v^{⊥}_{1}|=1}} ⟨v_{0}, v_{1}^{⊥}⟩ = sin θ(V_{1}^{⊥}, V_{0}^{⊥}).

It is important to observe that in general θ(V_{0}, V_{1}) = θ(V_{1}, V_{0}) does not hold: for instance if V_{0} ⊂ V_{1} we have θ(V_{0}, V_{1}) = 0 and θ(V_{1}, V_{0}) = π/2.

By applying this concept to a smooth curve γ(t) on M, with V_{t} = T_{γ(t)}M, we clearly have that both functions

(t, s) → θ(V_{t}, V_{s}),
(t, s) → θ(V_{s}, V_{t}),

are Lipschitz. Therefore for some constant C both the following inequalities hold:

(6.5a)  |u − P_{s}(u)| ≤ C|u||s − t|,  ∀t, s ∈ [0, 1] and u ∈ V_{t}.
(6.5b)  |P_{s}(u^{⊥})| ≤ C|u^{⊥}||s − t|,  ∀t, s ∈ [0, 1] and u^{⊥} ∈ V_{t}^{⊥}

The idea under the construction we are going to introduce is given by the following formula:

(6.6)  ∇_{v(0)}P_{t}(u) = 0, ∀u ∈ V_{0}.

That is: the vectors P_{t}(u) are a first order approximation of the parallel transport. Equation (6.6) follows by applying inequalities (6.5) (note that u − P_{t}(u) ∈ V_{t}^{⊥}):

|P_{0}(u − P_{t}(u))| ≤ Ct|u − P_{t}(u)| ≤ C^{2}t^{2}|u|.

Now let 𝔓 be the direct set of all the partitions of [0, 1], where P ≥ Q, P, Q ∈ 𝔓, if P is a refinement of Q. For P = {0 = t_{0} < t_{1} < ⋯ < t_{N} = 1} ∈ 𝔓 and u ∈ V_{0} define P(u) ∈ V_{1} as:

P(u) := P_{1}(P_{t_{N−1}}(⋯ P_{0}(u))).

Our first goal is to prove that the limit P(u) for P ∈ 𝔓 exists. After this is done, it will be naturally defined a curve t → u_{t} ∈ V_{t} by taking partitions of [0, t] instead of [0, 1]: the final target is to show that this curve is actually the parallel transport of u along the curve γ.

The proof is based on the following lemma.

Lemma 6.2. Let 0 ≤ s_{1} ≤ s_{2} ≤ s_{3} ≤ 1 be given numbers. Then it holds:

|P_{s_{3}}(u) − P_{s_{3}}(P_{s_{2}}(u))| ≤ C^{2}|u||s_{1} − s_{2}||s_{2} − s_{3}|, ∀u ∈ V_{s_{1}}.

Proof. Since P_{s_{3}}(u) − P_{s_{3}}(P_{s_{2}}(u)) = (P_{s_{3}}(Id − P_{s_{2}})(u)), the proof is a straightforward application of inequalities (6.5). □

From this lemma it follows easily by induction that the following inequality holds for any 0 ≤ s_{1} < ⋯ < s_{N} ≤ 1 and u ∈ V_{s_{1}}:

\begin{matrix}|P_{s_{N}} (u) − P_{s_{N}} (P_{s_{N−1}} & ⋯ (P_{s_{2}}(u)))| \\ ≤ & |P_{s_{N}} (u) − P_{s_{N}} (P_{s_{N−1}}(u))| + |P_{s_{N}} (P_{s_{N−1}}(u)) − P_{s_{N}} (P_{s_{N−1}} ⋯ (P_{s_{2}}(u)))| \\ ≤ & C^{2}|u||s_{1} − s_{N−1}||s_{N−1} − s_{N}| + |P_{s_{N−1}}(u) − P_{s_{N−1}} ⋯ (P_{s_{2}}(u))| \\ ≤ & ⋯ \\ ≤ & C^{2}|u| ∑_{i=2}^{N−1} |s_{1} − s_{i}||s_{i} − s_{i+1}| \\ ≤ & C^{2}|u||s_{1} − s_{N}|^{2}. \end{matrix} (6.7)

With this result, we can prove the existence of the limit of P (u) as P varies in 𝔓.

Theorem 6.3. For any u ∈ V_{0} there exists the limit of P(u) as P varies in 𝔓.

Proof. We have to prove that given ε > 0 there exists a partition P such that it holds

(6.8)  |P(u) − Q(u)| ≤ |u|ε, ∀Q ≥ P.

In order to do so it is sufficient to find 0 = t_{0} < t_{1} < ⋯ < t_{N} = 1 such that ∑_{i} |t_{i+1} − t_{i}|^{2} ≤ ε/C^{2}, and repetedly apply equation (6.7) (we will come back to this proof with more details in theorem 6.31). □

Now we can introduce the maps T_{s}^{t} : V_{s} → V_{t} which associate to the vector u ∈ V_{s} the limit of the process just described (taking into account partitions of [s, t]).

Theorem 6.4. For any t_{1} ≤ t_{2} ≤ t_{3} ∈ [0, 1] it holds
(6.9)  T_{t_{2}}^{t_{3}} ∘ T_{t_{1}}^{t_{2}} = T_{t_{1}}^{t_{3}}.

Moreover for any u ∈ V_{0} the curve t → u_{t} := T_{0}^{t}(u) ∈ V_{t} is the parallel transport of u along γ.

2. ON THE ANGLE BETWEEN TANGENT SPACES IN 𝒫_{2}(ℝ^{d}) 93

Proof. To prove equation (6.9) it is sufficient to consider those partitions of [t_{1}, t_{3}] which contain t_{2} and pass to the limit first on [t_{1}, t_{2}] and then on [t_{2}, t_{3}].

To prove the second part of the statement observe that due to (6.9) it is sufficient to check that the covariant derivative is 0 at 0. Note that from (6.7) it follows

|P_{t}(u) − u_{t}| ≤ C^{2}t^{2},

therefore the thesis follows from (6.6). □

2. On the angle between tangent spaces in 𝒫_{2}(ℝ^{d})

The construction we did on regular manifolds embedded in ℝ^{d} shows that the key step which allows to prove the existence of the parallel transport is the Lipschitz property of the angle between tangent spaces. In this section we introduce the analogous notion of angle for the space 𝒫_{2}(ℝ^{d}) and analyze its properties.

An important difference with the case of a manifold embedded in ℝ^{d}, is that in our context the two spaces Tan_{μ}(𝒫_{2}(ℝ^{d})) and Tan_{ν}(𝒫_{2}(ℝ^{d})) are not (affine) subspaces of a bigger Hilbert space, therefore we cannot directly imitate the definition of angle given in the first section. However, a natural way to embed L_{ν}^{2} into L_{μ}^{2} is given by the composition with a map T pushing μ into ν: to a map f ∈ L_{ν}^{2} we can associate the “translated” map f ∘ T ∈ L_{μ}^{2}.

Clearly the translation through a map is an isometry from L_{μ}^{2} to L_{ν}^{2}. The definition of angle comes out naturally.

Definition 6.5 (Angle between tangent spaces through a map). Let μ, ν ∈ 𝒫_{2}(ℝ^{d}) be two measures and let T be a transport map from μ to ν. Then the angle θ_{T} (μ, ν) ∈ [0, π/2] between the tangent spaces at μ and ν through the map T is given by

sin θ_{T} (μ, ν) := sup ∥P_{μ}^{⊥}(v ∘ T)∥_{μ},

where P_{μ}^{⊥} = Id − P_{μ} where is the orthogonal projection onto Tan_{μ}^{⊥}(𝒫_{2}(ℝ^{d})) and the supremum is taken among all v ∈ Tan_{ν}(𝒫_{2}(ℝ^{d})) such that ∥v∥_{ν} = 1.

It is important to note that the angle between the tangent spaces at two measures, strongly depends on the transport maps used.

Observe that, assuming that the transport map T is invertible, the angle θ_{T} (μ, ν) is in general not equal to the angle θ_{T^{−1}}(ν, μ): this reflects the fact that the are two angles between two subspaces V_{1} and V_{2} of ℝ^{d}, depending on whether we are considering projections from V_{1} onto V_{2} or from V_{2} to V_{1}.

The fundamental bound on the angle we are going to use in the rest of the chapter is given by the following proposition: the key requirement is the Lipschitz property of the transport map, while there is no regularity assumption for the measures involved.

From now on, for a given vector valued map T ∈ L_{μ}^{2} we will indicate with Lip(T) the infimum of the least Lipschitz constant among all the functions equal to T μ−a.e..

Proposition 6.6. Let μ, ν ∈ 𝒫_{2}(ℝ^{d}) and T ∈ L_{μ}^{2} a transport map from μ to ν. Suppose that T is (equivalent to) a Lipschitz function. Then it holds

(6.10)  sin θ_{T} (μ, ν) ≤ Lip(T − Id).

Proof. Equation (6.10) is equivalent to

(6.11)  ∥∇φ ∘ T − P_{μ}(∇φ ∘ T)∥_{μ} ≤ ∥∇φ∥_{ν} Lip(T − Id), ∀φ ∈ C_{c}^{∞}(ℝ^{d}, ℝ).

Let us suppose at first that T − Id ∈ C_{c}^{∞}(ℝ^{d}, ℝ^{d}). In this case φ ∘ T ∈ C_{c}^{∞}(ℝ^{d}), and therefore its gradient ∇(φ ∘ T) = ∇T · (∇φ) ∘ T belongs to Tan_{μ}(𝒫_{2}(ℝ^{d})). From the minimality properties of the projection we get:

∥∇φ ∘ T − P_{μ}(∇φ ∘ T)∥_{μ} ≤ ∥∇φ ∘ T − ∇T · (∇φ) ∘ T ∥_{μ}
= (∫ |(I − ∇T (x)) · ∇φ(T (x))|^{2}dμ(x))^{1/2}

≤ (∫ |∇φ(T (x))|^{2}∥∇(Id − T)(x)∥_{op}^{2}dμ(x))^{1/2}
≤ ∥∇φ∥_{ν} Lip(T − Id),

where I is the identity matrix and ∥∇(Id − T)(x)∥_{op} is the operator norm of the linear functional from ℝ^{d} to ℝ^{d} given by v → ∇(Id − T)(x) · v.

Now turn to the general case. Find a sequence (T_{n} − Id) ⊂ C_{c}^{∞}(ℝ^{d}, ℝ^{d}) such that

T_{n} → T  uniformly on compact sets,
\overline{lim}_{n} Lip(T_{n} − Id) ≤ Lip(T − Id).

It is clear that for such a sequence it holds ∥T − T_{n}∥_{μ} → 0, therefore we have

∥∇φ ∘ T − P_{μ}(∇φ ∘ T)∥_{μ} ≤ ∥∇φ ∘ T − ∇(φ ∘ T_{n})∥_{μ}
≤ ∥∇φ ∘ T − ∇φ ∘ T_{n}∥ + ∥∇φ ∘ T_{n} − ∇(φ ∘ T_{n})∥
≤ Lip(∇φ)∥T − T_{n}∥_{μ} + ∥∇φ ∘ T_{n}∥_{μ} Lip(T_{n} − Id).

Letting n → +∞ we get the thesis. □

In the rest of the section we study the infinitesimal behavior of the angle between two tangent spaces. The results proven, although not needed in the following, help understanding the behavior of tangent and normal vectors w.r.t. (weak) convergence of maps.

Start observing that the notion of translation through a map may be generalized to the case of translation through a plan: the only difference being that the translation through a plan is no more an isometry, but just a non-expansive map.

Definition 6.7 (Composition of a vector with a plan). For any couple of measures μ, ν ∈ 𝒫_{2}(ℝ^{d}) and any admissible plan γ ∈ Adm(μ, ν) we define the composition of a vector f ∈ L_{ν}^{2} with γ as the vector f ∘ γ ∈ L_{μ}^{2} defined by:

f ∘ γ(x) := ∫ f(y)dγ_{x}(y).

The inequality

∥f ∘ γ∥_{μ}^{2} = ∫ (∫ f(y)dγ_{x}(y))^{2} dμ(x) ≤ ∫ |f(y)|^{2}dγ(x, y) = ∥f∥_{ν}^{2},

shows that the composition with a plan is a non-expansive map. It is worth noticing that, letting α be the unique 3-plan satisfying

\begin{aligned}π_{#}^{1,2}α & = γ, \\ π_{#}^{2,3}α & = (Id, f)_{#}ν, \end{aligned}

the composition f ∘ γ is nothing but the barycentric projection of the marginal π_{#}^{1,3}α.

Remark 6.8. Actually there is no particular need to take the barycentric projection instead than simply the marginal, if one does so, he defines a norm preserving map from L_{ν}^{2} to 𝒫_{2}(ℝ^{2d})_{μ}. We preferred to take the projection since we are only interested in translation of maps, and therefore it is simpler to work only with maps, instead than with plans.

Remark 6.9. Obviously it is possible to define the composition of a plan in 𝒫_{2}(ℝ^{2d})_{ν} with a transport plan in Adm(μ, ν): it is what we already did a lot of times by using Dudley’s lemma. What comes out is a multi-valued operator continuous w.r.t. the Hausdorff distance between compact subsets of 𝒫_{2}(ℝ^{2d})_{μ}.

Definition 6.10 (Angle between tangent spaces through a plan). Let μ, ν ∈ 𝒫_{2}(ℝ^{d}) be two measures and γ ∈ Adm(μ, ν) an admissible plan. Then the angle θ_{γ}(μ, ν) ∈ [0, π/2] between the tangent spaces at μ and ν through the plan γ is defined as:

sin θ_{γ}(μ, ν) := sup ∥P_{μ}^{⊥}(v ∘ γ)∥_{μ}, v ∈ Tan_{ν}(𝒫_{2}(ℝ^{d})), ∥v∥_{ν} = 1.

As for the angles through maps, in general it holds θ_{γ}(μ, ν) ≠ θ_{γ}_{−1}(ν, μ).

We want to understand whether the angles θ_{γ}(μ, ν), θ_{γ}_{−1}(ν, μ) converge to 0 when μ is a fixed measure and γ converges to (Id, Id)_{#}μ.

Let (μ_{n}) be a sequence of measures W −converging to a measure μ, and γ_{n} ∈ Adm(μ, μ_{n}) such that ∥x_{1} − x_{2}∥_{γ_{n}} → 0 as n → ∞. In these hypotheses in general the angle θ_{γ_{n}}(μ, μ_{n}) does not converge to 0. Actually it can be proved something more, as the following proposition shows:

Proposition 6.11. Let μ ∈ 𝒫_{2}(ℝ^{d}) and u ∈ L_{μ}^{2}. Then there exists a sequence of measures (μ_{n}) and a sequence of maps u_{n} ∈ Tan_{μ_{n}}(𝒫_{2}(ℝ^{d})) such that (u_{n}) strongly converges to u.

Proof. Assume for a moment that μ ≪ L^{d} with positive density and that u is Lipschitz. For every n ∈ ℕ, define the “grid” G_{n} as the set of those points x in ℝ^{d} such that at least one coordinate of nx is integer, and the set H_{n} as

H_{n} := \{x ∈ ℝ^{d} : d(x, G_{n}) > \frac{1}{n^{2}} \} .

It is clear that μ(ℝ^{d} \ H_{n}) → 0 as n → ∞. Define

μ_{n} := \frac{1}{μ(H_{n})}μ|_{H_{n}}.

Now define, for every n ∈ ℕ, the function u_{n} ∈ L_{μ_{n}}^{2} as the function which is constant on each connected component of ℝ^{d} \ H_{n}, with value equal to the average of u on the connected component itself. Clearly u_{n} ∈ L_{μ_{n}}^{2}, moreover, since it is constant on each connected component, it holds u_{n} ∈ Tan_{μ_{n}}(𝒫_{2}(ℝ^{d})). Let L be the Lipschitz constant of u and conclude with the estimate

∥u_{n} − u∥_{μ_{n}} ≤ \frac{L\sqrt{d}}{n}.

For the general case, define μ_{ε} := μ ∗ ρ_{ε} and u_{ε} := (uμ) ∗ ρ_{ε}/μ_{ε}, approximate u_{ε} in L_{μ_{ε}}^{2} with Lipschitz functions and conclude with a diagonalization argument. □

Observe that the approximation of the measure μ we used, has the following properties: if μ ≪ L^{d} then μ_{n} ≪ L^{d}. Furthermore, in this case it holds \overline{lim} sup |μ_{n}/L^{d}| ≤ sup |μ/L^{d}|.

As said, this proposition implies that the angle \overline{θ_{γ_{n}}(μ, μ_{n})} doesn’t converge to 0 as n → ∞: indeed, observe that

sin θ_{γ}(μ, ν) = sup_{{v∈Tanν (𝒫_{2}(ℝ^{d})) \atop ∥v∥ν =1}} ∥P_{μ}^{⊥}(v ∘ γ)∥_{μ}
= sup_{{v∈Tanν (𝒫_{2}(ℝ^{d})) \atop ∥v∥ν =1}} sup_{{u∈Tan^{⊥}_{μ}(𝒫_{2}(ℝ^{d})) \atop ∥u∥μ=1}} ⟨u, v ∘ γ⟩_{μ}

= sup_{{v∈Tanν (𝒫_{2}(ℝ^{d})) \atop ∥v∥ν =1}} sup_{{u∈Tan^{⊥}_{μ}(𝒫_{2}(ℝ^{d})) \atop ∥u∥μ=1}} ∫ ⟨u(x), v(y)⟩dγ(x, y)

= sup_{{u∈Tan^{⊥}_{μ}(𝒫_{2}(ℝ^{d})) \atop ∥u∥μ=1}} sup_{{v∈Tanν (𝒫_{2}(ℝ^{d})) \atop ∥v∥ν =1}} ⟨u ∘ γ^{−1}, v⟩_{ν}

= sup_{{u∈Tan^{⊥}_{μ}(𝒫_{2}(ℝ^{d})) \atop ∥u∥μ=1}} ∥P_{ν}(u ∘ γ^{−1})∥_{ν}.

Now choose u ∈ Tan_{μ}^{⊥}(𝒫_{2}(ℝ^{d})) and find μ_{n} ∈ 𝒫_{2}(ℝ^{d}) , u_{n} ∈ Tan_{μ_{n}}(𝒫_{2}(ℝ^{d})) such that (u_{n}) strongly converges to u. Use remark 2.24 to get that for any choice of plans γ_{n} ∈ Adm(μ, μ_{n}) satisfying lim_{n} ∥x−y∥_{γ_{n}} = 0 the sequence (u∘γ_{n}^{−1}) strongly converges to u and it holds ∥u ∘ γ_{n}^{−1} − u_{n}∥_{γ_{n}} → 0, and conclude by the minimizing properties of the projection:

\underline{lim}_{n→∞} ∥P_{ν}(u ∘ γ_{n}^{−1})∥_{ν}^{2} = \underline{lim}_{n→∞} (∥u ∘ γ_{n}^{−1}∥_{ν}^{2} − ∥u ∘ γ_{n}^{−1} − P_{μ_{n}}(u ∘ γ_{n}^{−1})∥_{μ_{n}}^{2})

≥ lim_{n→∞} ∥u ∘ γ_{n}^{−1}∥_{ν}^{2} − \overline{lim}_{n→∞} ∥u ∘ γ_{n}^{−1} − u_{n}∥_{μ_{n}}^{2} = ∥u∥_{μ}^{2}.

With the same notation, regarding the angle θ_{γ}{−1 \atop n}(μ_{n}, μ) we don’t have a final word, however we can show that for any u ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) the vectors u∘γ_{n}^{−1} −P_{μ_{n}}(u∘γ_{n}^{−1}) strongly converge to 0.

Proposition 6.12. Let μ, μ_{n} ∈ 𝒫_{2}(ℝ^{d}), n ∈ ℕ, and γ_{n} ∈ Adm(μ, μ_{n}) weakly converging to (Id, Id)_{#}μ as n → ∞. Then

(6.12)  ∥u ∘ γ_{n}^{−1} − P_{μ_{n}}(u ∘ γ_{n}^{−1})∥_{μ_{n}} → 0, ∀u ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})).

Proof. Observe that by the uniform continuity of the operators u → u∘γ_{n}^{−1}−P_{ν}(u∘γ_{n}^{−1}) (their norm is less or equal to 1), it is sufficient to prove equation (6.12) for u = ∇φ as φ varies in C_{c}^{∞}(ℝ^{d}). Now notice that ∇φ ∈ Tan_{μ_{n}}(𝒫_{2}(ℝ^{d})) for every n, therefore by the minimizing property of the projection we have:

∥∇φ ∘ γ_{n}^{−1} − P_{ν}(∇φ ∘ γ_{n}^{−1})∥_{μ_{n}}^{2} ≤∥∇φ ∘ γ_{n}^{−1} − ∇φ∥_{μ_{n}}^{2}
≤ ∫_{supp(φ)^{2}} |∇φ(x_{1}) − ∇φ(x_{2})|^{2}dγ_{n}(x_{1}, x_{2})
≤ Lip(∇φ) ∫_{supp(φ)^{2}} |x_{1} − x_{2}|^{2}dγ_{n}(x_{1}, x_{2}),

and the last term goes to 0 as n → ∞ by the weak convergence of γ_{n} to (Id, Id)_{#}μ.

□

We can summarize the previous propositions by saying:

• A sequence of tangent vectors strongly converging, does not necessarily converge to a tangent vector, while a sequence of normal vectors weakly converging, converges to a normal vector

• For a given sequence (μ_{n}) converging to μ, every tangent vector in μ can be approximated by tangent vectors in μ_{n} w.r.t. the strong convergence of maps, while a normal vector may not be approximated by normal vector, not even w.r.t. the weak convergence of maps.

It is therefore interesting to know under which conditions on the measures, we have that any limit (possibly weak) of tangent vectors is a tangent vector. We don’t have a general answer on this, however proposition 6.6 and the proof of 4.14 give the sufficient conditions collected by the following proposition.

Proposition 6.13. Let (μ_{n}) be a sequence of measures weakly converging to a measure μ and u_{n} ∈ Tan_{μ_{n}}(𝒫_{2}(ℝ^{d})). Assume that (u_{n}) weakly converge to some u ∈ L_{μ}^{2} and that one of the two following conditions holds:

• there exists a sequence of transport maps (not necessarily optimal) (T_{n}) from μ to μ_{n} such that (Lip(T_{n} − Id)) converges to 0 as n goes to infinity,

• there exists a sequence (ρ_{n}) of regular measures weakly converging to δ_{0} such that μ_{n} = μ ∗ ρ_{n}.

Then u ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})).

Proof. Suppose the first condition is true. Then from proposition 6.6 we have ∥u_{n} ∘ T_{n} − P_{μ}(u_{n} ∘ T_{n})∥_{μ} ≤ Lip(T_{n} − Id)∥u_{n}∥_{μ_{n}} therefore the sequence (P_{μ}(u_{n} ∘ T_{n})) weakly converges to u, too. Since P_{μ}(u_{n} ∘ T_{n}) ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) we get the conclusion.

Now assume the second condition is true and choose a vector w ∈ Tan_{μ}^{⊥}(𝒫_{2}(ℝ^{d})): we will show that ⟨w, u⟩_{μ} = 0. By the Radon-Nikodym derivation theorem and the absolute continuity of (wμ) ∗ ρ_{n} w.r.t. μ_{n} = μ ∗ ρ_{n} we know that the function

w_{n} := \frac{(wμ) ∗ ρ_{n}}{μ_{n}} ,

is well defined for any n ∈ ℕ. Moreover from lemma 3.3 we know that ∥w_{n}∥_{μ_{n}} ≤ ∥w∥_{μ}. It is clear that (w_{n}) strongly converges to w and that ∇ · (w_{n}μ_{n}) = 0, therefore using lemma 2.28 we can pass to the limit in ⟨u_{n}, w_{n}⟩_{μ_{n}} = 0 to get ⟨u, w⟩_{μ} = 0 as claimed.

□

3. Regular curves

In this section we will introduce the regular curves, which is a dense class of curves, along which a parallel transport can be defined and exists.

By analogy with the classical Riemannian case, (see equation (6.4)), and due to the discussion of the preceding section, we wish to say that u_{t} is a parallel transport along a curve μ_{t} if

lim_{h→0} P_{ρ_{t}} (\frac{u_{t+h} ∘ T(t, t + h, ·) − u_{t}}{h} ) = 0, ∀t ∈ [0, T],

for a suitable class of maps T(t, s, ·) satisfying T(t, s, ·)_{#}μ_{t} = μ_{s}. Given that we want to apply proposition 6.6 (to replicate the approximation scheme for the construction of parallel transport), it is natural to ask for the maps T(t, s, ·) to be Lipschitz. From a technical point of view, it seems necessary that T(t, s, ·) have the semi-group property, i.e. T(t, s, T(r, t, ·)) = T(r, s, ·) (see the proof of lemma 6.29 on the next section). Finally, in order to have the parallel transport to induce the Levi-Civita connection on (𝒫_{2}(ℝ^{d}), W), it must hold

\frac{d}{ds}T(t, s, x)|_{s=\overline{s}} = v\overline{s}(T(t, \overline{s}, x)), ∀t, \overline{s} ∈ [0, 1], x ∈ supp(μ_{t}),

where v_{t} are the velocity vectors of μ_{t} (see the discussion in section 5 for a further explanation).

Since we know from the classical Cauchy-Lipschitz theory that whenever the velocity vectors v_{t} of ρ_{t} satisfy

∫_{0}^{1} Lip(v_{t})dt < +∞,

then the flow maps exist and are Lipschitz functions of the space variable, we give the following definition:

Definition 6.14 (Regular curves). Let [0, T] ∋ t → μ_{t} ∈ 𝒫_{2}(ℝ^{d}) be an absolutely continuous curve and let v_{t} be its velocity vector field defined, for a.e. t ∈ [0, T], by

\begin{matrix}v_{t} ∈ Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})), \\ \frac{d}{dt}μ_{t} + ∇ · (v_{t}μ_{t}) = 0. \end{matrix}

We say that μ_{t} is regular if it holds

∫_{0}^{T} Lip(v_{t})dt < +∞.

Observe that we are making no regularity assumption on the measures μ_{t}.

It is worth noticing that the definition of regularity for a curve is independent from its (Lipschitz) parameterization. In the following we will always assume that a regular curve is parameterized in [0, 1].

As already pointed out, for a given regular curve it exists a unique family of maps T(t, s, x) : [0, 1] × [0, 1] × ℝ^{d} → ℝ^{d}, which we call the flow of the curve, absolutely continuous in s and Lipschitz in x satisfying

(6.13)  \{ \begin{matrix}T(t, t, x) = x, & ∀x, t, \\ \frac{d}{ds}T(t, s, x)|_{s=\overline{s}} = v\overline{s}(T(t, \overline{s}, x)) & ∀x, t, a.e. \overline{s}, \\ T(s, r, T(t, s, x)) = T(t, r, x), & ∀x, t, s, r \\ (T(t, s, x))_{#}μ_{t} = μ_{s} & ∀t, s ∈ [0, 1]. \end{matrix}

Observe that we have the following bound on the Lipschitz constant of T(t, s, ·)−Id:

(6.14)  Lip(T(t, s, ·) − Id) ≤ exp (|∫_{t}^{s} Lip(v_{r})dr|) − 1, t, s ∈ [0, 1].

This inequality is a well known consequence of the validity of equations (6.13), we recall the proof for the sake of completeness.

Proposition 6.15. Let T(t, s, ·) be the flow maps of a regular curve. Then the following estimates hold:

Lip(T(t, s, ·)) ≤ exp (|∫_{t}^{s} Lip(v_{r})dr|), t, s ∈ [0, 1],
Lip(T(t, s, ·) − Id) ≤ exp (|∫_{t}^{s} Lip(v_{r})dr|) − 1, t, s ∈ [0, 1].

Proof. The first equation follows by a direct application of Gronwall lemma to the inequality

\frac{d}{ds}|T(t, s, x) − T(t, s, y)|^{2} =2⟨T(t, s, x) − T(t, s, y), v_{s}(T(t, s, x)) − v_{s}(T(t, s, y))⟩
≤2|T(t, s, x) − T(t, s, y)|^{2} Lip(v_{s}).

For the second one, observe that it holds

\frac{d}{ds}|T(t, s, x) − x − T(t, s, y) + y|^{2}
=2⟨T(t, s, x) − x − T(t, s, y) + y, v_{s}(T(t, s, x)) − v_{s}(T(t, s, y))⟩
≤2|T(t, s, x) − x − T(t, s, y) + y∥x − y| Lip(v_{s}) Lip(T(t, s, ·)).

therefore the conclusion follows by integrating from t to s the inequality

\frac{d}{ds}|T(t, s, x) − x − T(t, s, y) + y| ≤ |x − y| Lip(v_{s}) exp (|∫_{t}^{s} Lip(v_{r})dr|) .

□

For a vector field t → u_{t} ∈ L_{μ_{t}}^{2} along a regular curve, we give the following definition of absolute continuity.

Definition 6.16 (Absolutely continuous vector fields). Let t → μ_{t} be a regular curve and t → u_{t} ∈ L_{μ_{t}}^{2} a vector field along it. We say that t → u_{t} is absolutely continuous if the maps t → u_{t} ∘ T(s, t, ·) ∈ L_{μ_{s}}^{2} are absolutely continuous for any s ∈ [0, 1], where T(t, s, ·) are the flow maps of μ_{t}. For an absolutely continuous vector field we will write \frac{d}{dt}u_{t} ∈ L_{μ_{t}}^{2} for its derivative defined by:

\frac{d}{dt}u_{t} := lim_{h→0} \frac{u_{t+h} ∘ T(t, t + h, ·) − u_{t}}{h} = \frac{d}{dt}(u_{t} ∘ T(s, t, ·)) ∘ T(t, s, ·), ∀s ∈ [0, 1].

Given that the composition with T(t, s, ·) is an isometry from L_{μ_{s}}^{2} to L_{μ_{t}}^{2}, it is clear that a vector field t → u_{t} is absolutely continuous if and only if for some s ∈ [0, 1] the curve t → u_{t} ∘ T(s, t, ·) ∈ L_{μ_{s}}^{2} is absolutely continuous. A simple consequence of the definition is that for an absolutely continuous vector field u_{t}, the map t → ∥u_{t}∥_{μ_{t}} is absolutely continuous.

It is important to underline that the definition of derivative of an absolutely continuous vector field allows us to take derivative of a function t → u_{t} whose range belongs to different L^{2} spaces as t varies and that this derivative is not the partial derivative of some function of the kind (t, x) → u_{t}(x). To see this, consider a case in which the measures μ_{t} are mutually singular (this happens, for instance, if the measure μ_{t} is concentrated on some k−dimensional submanifold of ℝ^{d}, k < d, which moves in time): in this case the function u_{t} is not defined on the support of μ_{s} for s ≠ t. Nevertheless, for regular curves, the above provides a good definition of derivative of a vector field along the curve.

In the following we will extensively use, without explicit mention, the fact that for any φ ∈ C_{c}^{∞}(ℝ^{d}, ℝ), the vector field t → ∇φ ∈ L_{μ_{t}}^{2} is tangent and absolutely continuous. Its derivative in the sense of the above definition is ∇^{2}φ · v_{t}.

Observe that the definition of derivative of an absolutely continuous vector field is consistent with the Leibnitz rule, as shown by the following calculation:

\begin{aligned}\frac{d}{dt}⟨u_{t}^{1}, u_{t}^{2}⟩_{μ_{t}} & = \frac{d}{dt}⟨u_{t}^{1} ∘ T(0, t, ·), u_{t}^{2} ∘ T(0, t, ·)⟩_{μ_{0}} \\ & = ⟨\frac{d}{dt}(u_{t}^{1} ∘ T(0, t, ·)), u_{t}^{2} ∘ T(0, t, ·)⟩_{μ_{0}} + ⟨u_{t}^{1} ∘ T(0, t, ·), \frac{d}{dt}(u_{t}^{2} ∘ T(0, t, ·))⟩_{μ_{0}} \\ & = ⟨\frac{d}{dt}u_{t}^{1}, u_{t}^{2}⟩_{μ_{t}} + ⟨u_{t}^{1}, \frac{d}{dt}u_{t}^{2}⟩_{μ_{t}} \end{aligned} (6.15)

We conclude the section by analyzing the first properties of regular curves: topological constraints, stability and density in the class of absolutely continuous curves. This part of the work may be skipped in a first reading: if the reader is more interested in the construction of the parallel transport along regular curves, we suggest to go directly to section 4.

Remark 6.17. An important consequence of the proposition 6.15, is a strong topological restriction on the supports of the measures μ_{t} along a regular curve. Indeed, from the fact that the flow maps T(t, s, ·) : ℝ^{d} → ℝ^{d} are (actually, can be extended to) bi-Lipschitz homeomorphisms of ℝ^{d} into itself and from supp(μ_{s}) = T(t, s, supp μ_{t}), we obtain that the supports of the measures μ_{t} are all homeomorphic.

In particular, if supp μ_{0} has n connected components, say A_{0}^{1}, …, A_{0}^{n}, then A_{t}^{i} := T(t, 0, ·)(A_{0}^{i}), i = 1, …, n, are the connected components of supp(μ_{t}) and it holds

μ_{t}|_{A_{t}^{i}} = T(t, 0, ·)_{#}(μ_{0}|_{A_{0}^{i}}), i = 1, …, n.

It follows that the mass μ_{t}(A_{t}^{i}) of each connected component is constant in time. Furthermore, it is easy to check that the velocity vectors of μ_{t}^{i} are (the restriction of) the vectors v_{t}, thus the curves t → μ_{t}^{i} := μ_{0}(A_{i})^{−1}μ_{t}|_{A_{t}^{i}}, i = 1, …, n are regular, too,

and their flow maps are (the restriction of) the flow maps of μ_{t}. Notice that from the definition of absolutely continuous vector field, it is easy to check that a vector field t → u_{t} ∈ L_{μ_{t}}^{2} is absolutely continuous if and only if so are the vector fields t → u_{t} ∈ L_{μ_{t}^{i}}^{2}, i = 1, …, n.

We want now to point out a stability result for regular curves. This result is a direct consequence of theorem 3.5 of [12], which gives sufficient conditions for the flow maps of a sequence of curves (μ_{t}^{n}) to converge to the flow maps of the limit curve μ_{t}. Here we skip the proof. Observe that the hypothesis we make are slightly weaker than those of 3.5 of [12], given that we only assume convergence w.r.t. the weak topology of 𝒫_{2}(ℝ^{d}), however it is easy to check, following the same proof, that the result is still valid.

Theorem 6.18 (Stability of regularity). Let (μ_{t}^{n}) be a sequence of regular curves such that for every t the sequence (μ_{t}^{n}) converges to some μ_{t} w.r.t. the weak topology of 𝒫_{2}(ℝ^{d}). Assume also that the limit curve t → μ_{t} is absolutely continuous, let v_{t}^{n}, v_{t} be the velocity vector fields of μ_{t}^{n}, μ_{t} respectively, and suppose that

\begin{aligned}\overline{lim}_{n→∞} ∥v_{t}^{n}∥_{μ_{t}^{n}} & ≤ ∥v_{t}∥_{μ_{t}}, a.e. t ∈ [0, 1], \\ Lip(v_{t}^{n}) & ≤ g(t), ∀n ∈ ℕ, a.e. t ∈ [0, 1], \end{aligned}

for some function g ∈ L^{1}(0, 1).

Then it holds Lip(v_{t}) ≤ g(t) for a.e. t, thus the limit curve μ_{t} is regular. Furthermore for a.e. t the sequence (v_{t}^{n}) strongly converges to v_{t} and for every t, s ∈ [0, 1] the flow maps T^{n}(t, s, ·) of μ_{t}^{n} strongly converge to T(t, s, ·).

We conclude this section with a proof of a density result for regular curves. It is well-known that the set

𝒫_{2}^{a}(ℝ^{d}) := {μ ∈ 𝒫_{2}(ℝ^{d}) : μ ≪ ℒ^{d}}

is a geodesic subspace of 𝒫_{2}(ℝ^{d}) (i.e. any geodesic between two points in 𝒫_{2}^{a}(ℝ^{d}) is entirely contained in 𝒫_{2}^{a}(ℝ^{d})) and the same is true for the subsets {μ = ρℒ^{d} :

∥ρ∥_{∞} ≤ C}. Our approximation will be obtained with measures in this class, and preserves these upper bounds on the densities, if any.

The delicate point in the approximation result is due to the fact that regularity imposes a Lipschitz condition on the tangent velocity field. The typical approximation schemes for solutions to the continuity equation, on the other hand, produce a regularized vector field that is compatible with the regularized density, but it is not tangent in general. Therefore a further projection of the regularized velocity on the tangent space is needed.

The following lemma will be used in the reduction to compactly supported measures.

Lemma 6.19 (Monotone approximation). Let μ_{t} : [0, 1] → 𝒫_{2}(ℝ^{d}) be absolutely continuous and let v_{t} be its tangent velocity field. Then there exist absolutely continuous curves μ_{t}^{n} : [0, 1] → 𝒫_{2}(ℝ^{d}) and z_{n} ↑ 1 satisfying:

(i) z_{n}μ_{t}^{n} ↑ μ_{t} for all t ∈ [0, 1] and sup_{t} W_{2}(μ_{t}^{n}, μ_{t}) → 0 as n → ∞; (ii) the tangent velocity field of μ_{t}^{n} is v_{t}, and there exists a closed ball B_{n} such that

supp μ_{t}^{n} ⊂ B_{n} for all t ∈ [0, 1].

Proof. Let Ω be the Banach space of continuous maps from [0, 1] to ℝ^{d} and let e_{t} : Ω → ℝ^{d} be the evaluation maps at time t, i.e. e_{t}(ω) = ω(t). According to [11, 8.2.1], we can represent μ_{t} as the law under e_{t} of a suitable probability measure η in Ω, concentrated in the set of absolutely continuous solutions of the equation ω̇ = v_{t}(ω).

Let

Ω_{n} := {ω ∈ Ω : ω is absolutely continuous, |ω(0)| ≤ n, ∫_{0}^{1} |ω̇|dt ≤ n}

and set η_{n} = χ_{Ω_{n}}η, z_{n} = η(Ω_{n}) = η_{n}(Ω) and μ_{t}^{n} = z_{n}^{−1}(e_{t})_{#}η_{n}. It is easy to check condition (i), and that the support of μ_{t}^{n} is contained in the ball \overline{B}_{2n}(0). Since also η_{n} is concentrated on curves solving the ODE ω̇ = v_{t}(ω), it turns out that v_{t} is an admissible velocity field for μ_{t}^{n} (i.e. the continuity equation holds, see again [11, 8.2.1] for instance). We conclude that v_{t} is the tangent velocity fields noticing that, because of condition (i), v_{t} ∈ Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})) implies v_{t} ∈ Tan_{μ_{t}^{n}}(𝒫_{2}(ℝ^{d})). □

We can now state our approximation result.

Theorem 6.20 (Approximation by regular curves). Let μ_{t} : [0, 1] → 𝒫_{2}(ℝ^{d}) be an absolutely continuous curve. Then there exist regular curves μ_{t}^{n} : [0, 1] → 𝒫_{2}(ℝ^{d}) satisfying:

(i) sup_{t} W_{2}(μ_{t}^{n}, μ_{t}) → 0 as n → +∞;

(ii) μ_{t}^{n} = ρ_{t}^{n}ℒ^{n}, sup_{t} ∥ρ_{t}^{n}∥_{∞} < +∞, ρ_{t}^{n} are smooth, the smooth tangent velocity fields v_{t}^{n} are gradients of smooth maps φ_{t}^{n} : ℝ^{d} → ℝ satisfying sup_{t} Lip(v_{t}^{n}) < ∞ and {ρ_{t}^{n} > 0} is a bounded open set with a smooth boundary;

(iii) if v_{t} ∈ L_{μ_{t}}^{2} is the tangent field of μ_{t}, we have that v_{t}^{n}μ_{t}^{n} weakly converge to v_{t}μ_{t} and

lim_{n→∞} ∫ |v_{t}^{n}|^{2}dμ_{t}^{n} = ∫ |v_{t}|^{2}dμ_{t}.

Proof. Step 1. Regularization of 1-periodic solutions.

By Lemma 6.19 and a diagonal argument we can assume that the supports of μ_{t} are contained in a fixed compact set. By a scaling argument, we can also assume with no loss of generality that the union of these supports is a compact subset K of (0, 1)^{d}. We consider the 1-periodic extension μ_{t}^{per} of μ_{t}, still solving the continuity equation with the 1-periodic extension v_{t}^{per} of v_{t}, and the regularized densities

ϱ_{t}^{n} := μ_{t}^{per} ∗ χ_{n},

still 1-periodic. Here χ_{n} is a family of smooth and symmetric w.r.t. 0, convolution kernels converging to δ_{0} whose support has a diameter equal to 2\sqrt{d}. With this choice of χ_{n}, we have inf_{t} inf ϱ_{t}^{n} > 0, and standard properties of convolution yield

sup ϱ_{t}^{n} ≤ sup_{t} μ_{t}^{per}([−1, 2]^{d}) sup χ_{n} = 3^{d} sup χ_{n}.

Analogous bounds hold, of course, for all higher order derivatives of ϱ_{t}^{n}. Passing to the velocity fields, we consider as in [36, 11] this regularization:

w_{t}^{n} := \frac{(v_{t}^{per}μ_{t}^{per}) ∗ χ_{n}}{ϱ_{t}^{n}}

which satisfies, thanks to the lower bound on ϱ_{t}^{n}, sup_{t} Lip(w_{t}^{n}) < ∞ (and the same holds for higher order derivatives) and preserves the validity of the continuity equation. Eventually we consider the projection v_{t}^{n} = ∇φ_{t}^{n} of w_{t}^{n} on periodic gradients by solving the PDE

∇ · (∇φ_{t}^{n}ϱ_{t}^{n}) = ∇ · (w_{t}^{n}ϱ_{t}^{n}).

From the variational formulation of the PDE we obtain

(6.16)  ∫_{(0,1)^{d}} |v_{n}^{t}|^{2}ϱ_{t}^{n} dx ≤ ∫_{(0,1)^{d}} |w_{n}|^{2}ϱ_{t}^{n}dx.

We can use standard elliptic regularity theory to obtain that sup_{t} Lip(v_{t}^{n}) < ∞. Moreover, using Jensen’s inequality as in [11, 8.1.10], we have the local estimate

(6.17)  ∫_{(0,1)^{d}} |w_{t}^{n}|^{2}ϱ_{t}^{n}dx ≤ ∫ ∫_{(0,1)^{d}} |v_{t}|^{2}(y)χ_{n}(x − y)dxdμ_{t}^{per}(y).

Step 2. Construction of the approximating sequence.

We build μ_{t}^{n} = ρ_{t}^{n}ℒ^{d} ∈ 𝒫_{2}^{a}(ℝ^{d}), with the same velocity field v_{t}^{n}, from the periodic measures ϱ_{t}^{n}ℒ^{d}. To this aim, we shall first consider ϱ_{t}^{n} as measures in the flat ddimensional torus 𝕋^{d} ∼ (0, 1)^{d}, with velocity field v_{t}^{n}. We denote by ℙ the Lebesgue measure on 𝕋^{d}, by X^{n}(t, x) : [0, 1] × 𝕋^{d} → 𝕋^{d} the smooth flow of v_{t}^{n} (starting from s = 0), and by η_{n} the probability measure in C([0, 1]; 𝕋^{d}) defined by

η_{n} := X^{n}(t, ·)_{#}(ϱ_{0}^{n}ℙ).

Equivalently, η_{n} is the law of the random variable x ↦ X^{n}(·, x) ∈ C([0, 1]; 𝕋^{d}) under ϱ_{0}^{n}ℙ. Classical representation results for solutions to the continuity equation with a Lipschitz vector field ensure that ϱ_{t}^{n}ℙ = X^{n}(t, ·)_{#}(ϱ_{0}^{n}ℙ), and since e_{t} ∘ X^{n}(·, x) = X^{n}(t, x) we obtain

(6.18)  (e_{t})_{#}η_{n} = ϱ_{t}^{n}ℙ  ∀t ∈ [0, 1], n ∈ ℕ.

From (6.16) and (6.17) we get

sup_{n} ∫_{0}^{1} ∫_{𝕋^{d}} |v_{t}^{n}|^{2}ϱ_{t}^{n}dℙ(x)dt < ∞

and this, using Prokhorov theorem as in [12, 44], gives that (η_{n}) is a relatively compact sequence in 𝒫(C([0, 1]; 𝕋^{d})). It is not restrictive, extracting if necessary a subsequence, to assume that (η_{n}) weakly converges, in the duality with continuous and bounded functions in C([0, 1]; 𝕋^{d}), to some probability measure η. Passing to the limit as n → ∞ in (6.18) we obtain that (e_{t})_{#}η = μ_{t}^{per} for all t ∈ [0, 1], and this means that η-almost all the paths ω are contained in K̃ (here we denote by K̃ the image of K in 𝕋^{d} and we consider μ_{t}^{per} as probability measures in 𝕋^{d}).

Now, let δ < 1 be such that K is contained in the interior of [δ, 1 − δ]^{d} and define η̃_{n} := z_{n}^{−1}χ_{Ω(δ)}η_{n}, where

Ω(δ) := {ω ∈ C([0, 1]; 𝕋^{d}) : ω(t) mod(1) ∈ (δ, 1 − δ)^{d} ∀t ∈ [0, 1]} , z_{n} := η_{n}(Ω(δ))

(in other words, we remove the trajectories that cross ∂(δ, 1−δ)^{d}). Since η is supported on paths contained in K̃, we have that z_{n} → 1 and still η̃_{n} weakly converge to χ_{Ω(δ)}η = η. We define

μ_{t}^{n} := (ẽ_{t})_{#}η̃_{n}

where ẽ_{t}(ω(t)) = ω_{t} mod(1) ∈ [0, 1)^{d}. The measures μ_{t}^{n} can also be represented by

(6.19)  μ_{t}^{n} = z_{n}^{−1}Y^{n}(t, ·)_{#}(χ_{E^{n}(δ)}ϱ_{0}^{n}ℒ^{d}),

where Y^{n}(t, x) = X^{n}(t, x) mod(1) and E^{n}(δ) = {x ∈ (0, 1)^{d} : X^{n}(·, x) ∈ Ω(δ)}.

By construction μ_{t}^{n} are probability measures in ℝ^{d} concentrated on [δ, 1 − δ]^{d}. It is immediate to check that the tangent field to μ_{t}^{n} is v_{t}^{n} (because η_{n} is concentrated on solutions to the ODE ω̇ = v_{t}^{n}(ω) in 𝕋^{d}, and v_{t}^{n} are gradients). In particular μ_{t}^{n} are regular curves and the convergence of μ_{t}^{n} to μ_{t} follows at once from the convergence of η̃_{n} to η, using the evaluation map ẽ_{t}. Notice also that the inequality z_{n}η̃_{n} ≤ η_{n} and the fact that the mass of their difference is infinitesimal imply

(6.20)  z_{n}μ_{t}^{n} ≤ ϱ_{t}^{n}ℒ^{d} and lim_{n→∞}(ϱ_{t}^{n}ℒ^{d} − z_{n}μ_{t}^{n})((0, 1)^{d}) = 0.

Step 3. Convergence of velocity fields.

Notice first that (6.16) and (6.17) give

lim sup_{n→∞} ∫ |v_{t}^{n}|^{2}dμ_{t}^{n} ≤ lim sup_{n→∞} ∫ |v_{t}^{n}(e_{t}(ω))|^{2}dη_{n}(ω) = lim sup_{n→∞} ∫ |v_{t}^{n}|^{2}ϱ_{t}^{n}dℙ
≤ lim sup_{n→∞} ∫ |w_{t}^{n}|^{2}ϱ_{t}^{n}dℙ
≤ lim sup_{n→∞} ∫ ∫_{(0,1)^{d}} |v_{t}|^{2}(y)χ_{n}(x − y)dxdμ_{t}^{per}(y) = ∫ |v_{t}|^{2}dμ_{t}.

Now, recall (see for instance [11, 9.4.3]) that the functional

G(ν, μ) := \begin{cases}∫ |f|^{2}dμ & if ν = fμ with f ∈ L^{2}(μ; ℝ^{d}), \\ +∞ & otherwise \end{cases}

is jointly lower semicontinuous in 𝒫_{2}(ℝ^{d}) × 𝒫_{2}(ℝ^{d}) with respect to weak convergence in the duality with C_{b}(ℝ^{d}), to obtain that any weak limit point σ of v_{t}^{n}μ_{t}^{n} as n → ∞ has the form ṽμ_{t} for some ṽ ∈ L_{μ_{t}}^{2} with ∥ṽ∥_{μ_{t}} ≤ ∥v_{t}∥_{μ_{t}}. On the other hand, passing to the limit in ∇ · ((w_{t}^{n} − v_{t}^{n})ϱ_{t}^{n}) = 0 and taking into account the weak convergence in the duality with C_{c}(ℝ^{d}) of w_{t}^{n}ϱ_{t}^{n} = (v_{t}^{per}μ_{t}^{per}) ∗ χ_{n} to v_{t}^{per}μ_{t}^{per} and the convergence to 0 in (0, 1)^{d} of ϱ_{t}^{n}ℒ^{d} − μ_{t}^{n} (ensured, even in the strong sense, by (6.20)) we get ∇ · ((ṽ − v_{t})μ_{t}) = 0. Since v_{t} is tangent and ∥ṽ∥_{μ_{t}} ≤ ∥v_{t}∥_{μ_{t}}, it must be ṽ = v_{t}. This proves the weak convergence of velocity fields that provides also, thanks to the lower semicontinuity of G, the lim inf inequality

lim inf_{n→∞} ∫ |v_{t}^{n}|^{2}dμ_{t}^{n} ≥ ∫ |v_{t}|^{2}dμ_{t}.

Step 4. Eventually we can regularize the characteristic function of the set E^{n}(δ) in (6.19), by smooth functions χ^{n} such that {χ_{n}ϱ_{0}^{n} > 0} is smooth and bounded, to approximate the curve μ_{t}^{n} by curves with the same velocity field and smooth densities with respect to ℒ^{d}, with smooth supports. □

The following is a useful technical result, which allows to approximate a regular curve with regular curves with range in the set of measures with finite support. The advantage of working with these measures μ, is that in this case an absolutely continuous curve is automatically regular: indeed, by continuity and compactness it is clear that inf_{t} inf_{x_{t}≠y_{t}} |x_{t} − y_{t}| > 0, where x_{t}, y_{t} ∈ supp μ_{t}, therefore, since the velocity vectors v_{t} are

defined only in a finite set, it is clear that t → Lip(v_{t}) is integrable as soon as t → ∥v_{t}∥_{μ_{t}} is.

Proposition 6.21. Let t → μ_{t} be a regular curve. Then there exists a sequence of regular curves (μ_{t}^{n}) such that for every n, t the support of μ_{t}^{n} is a finite set of points and it holds

lim_{n→∞} sup_{t∈[0,1]} W (μ_{t}^{n}, μ_{t}) = 0,

lim_{n→∞} ∥v_{t}^{n}∥_{μ_{t}^{n}} = ∥v_{t}∥_{μ_{t}}, a.e. t ∈ [0, 1],

sup_{n∈ℕ}Lip(v_{t}^{n}) ≤ Lip(v_{t}), a.e. t ∈ [0, 1],

where v_{t}, v_{t}^{n} are the velocity vectors of t → μ_{t} and t → μ_{t}^{n} respectively.

It is clear that the cardinality of the support of the measures of a regular curve, cannot change in time. Therefore the bound on the cardinality of supp(μ_{t}^{n}) is, for every n, uniform in t.

Proof. Let T(t, s, x) be the flow maps of μ_{t}. Find a sequence (μ_{0}^{n}) of measures with finite support such that supp(μ_{0}^{n}) ⊂ supp μ_{0} and W (μ_{0}, μ_{0}^{n}) → 0 as n → ∞. Now define

μ_{t}^{n} := T(0, t, ·)_{#}μ_{0}^{n}.

It is clear that t → μ_{t}^{n} is an absolutely continuous curve whose velocity vector field is v_{t}. Therefore the thesis will follow if we show the uniform convergence of t → μ_{t}^{n} to t → μ_{t}. Observe that the maps T(0, t, ·) are equilipschitz in t, let L be a common Lipschitz constant, choose a plan γ_{0}^{n} ∈ Opt(μ_{0}, μ_{0}^{n}) and define

γ_{t}^{n} := (T(0, t, ·), T(0, t, ·))_{#}γ_{0}^{n}.

It holds γ_{t}^{n} ∈ Adm(μ_{t}, μ_{t}^{n}), therefore

W (μ_{t}^{n}, μ_{t}) ≤ ∥x_{1}−x_{2}∥γ_{t}^{n} = ∥T(0, t, x_{1})−T(0, t, x_{2})∥γ_{0}^{n} ≤ L∥x_{1}−x_{2}∥γ_{0}^{n} = LW (μ_{0}^{n}, μ_{0}).

□

4. Parallel transport

In this section we define the parallel transport along regular curves in 𝒫_{2}(ℝ^{d}), show its existence, uniqueness and stability. We will also show by giving an explicit example that the lack of regularity may lead to the non-existence of the parallel transport, even along geodesics. At the end of the section we collect some examples of regular curves and parallel transports along them.

Definition 6.22 (Parallel transport along regular curves). Let μ_{t} be a regular curve, T(t, s, x) its flow and t → u_{t} ∈ Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})) a tangent vector field defined along the curve. We say that t → u_{t} is a parallel transport if it is absolutely continuous and it holds

(6.21)  P_{μ_{t}} (\frac{d}{dt}u_{t}) = 0, a.e. t ∈ [0, 1].

Equation (6.21) may be equivalently written as:

lim_{h→0} P_{μ_{t}} (\frac{u_{t+h} ∘ T(t, t + h, ·) − u_{t}}{h} ) = 0, in L_{μ_{t}}^{2}a.e. t ∈ [0, 1].

Another equivalent characterization, thanks to (6.15), is

(6.22) \frac{d}{dt}⟨∇η, u_{t}⟩_{μ_{t}} = ⟨∇^{2}η · v_{t}, u_{t}⟩_{μ_{t}}  for a.e. t ∈ (0, 1), for all η ∈ C_{c}^{∞}(ℝ^{d}),

or, in integral form:

(6.23)  ⟨∇η, u_{s}⟩_{μ_{s}} − ⟨∇η, u_{t}⟩_{μ_{t}} = ∫_{t}^{s}⟨∇^{2}η · v_{r}, u_{r}⟩_{μ_{r}}dr, ∀η ∈ C_{c}^{∞}.

Observe that this equation makes sense even if the underlying curve t ↦ μ_{t} is not regular, but only absolutely continuous. We will come back to this point in the last section.

It is easy to check that also the concept of parallel transport is invariant under reparameterization: if μ_{t} is a regular curve, u_{t} is a parallel transport along it and r : [0, R] → [0, 1] is a Lipschitz reparameterization of [0, 1], then μ̃_{s} := μ_{r(s)} is regular in [0, R] and ũ_{s} := u_{r(s)} is a parallel transport along it.

Proposition 6.23 (Linearity and conservation of scalar product). Let u_{t}^{1}, u_{t}^{2} be two parallel transports along a regular curve t → μ_{t} on [0, 1] and λ^{1}, λ^{2} ∈ ℝ. Then t → ⟨u_{t}^{1}, u_{r}^{2}⟩_{μ_{t}} is constant and t → λ^{1}u_{t}^{1} + λ^{2}u_{t}^{2} is a parallel transport.

Proof. The claim on λ^{1}u_{t}^{1} + λ^{2}u_{t}^{2} follows directly by the linearity of the definition of absolutely continuous vector field and of equation (6.21).

To prove that the scalar product is constant, just observe that t → ⟨u_{t}^{1}, u_{t}^{2}⟩_{μ_{t}} = \frac{1}{2}(∥u_{t}^{1}∥_{μ_{t}}^{2} + ∥u_{t}^{2}∥_{μ_{t}}^{2} − ∥u_{t}^{1} − u_{t}^{2}∥_{μ_{t}}^{2}) is absolutely continuous and that by the Leibnitz rule (6.15) its derivative is given by

\frac{d}{dt}⟨u_{t}^{1}, u_{t}^{2}⟩_{μ_{t}} = ⟨\frac{d}{dt}u_{t}^{1}, u_{t}^{2}⟩_{μ_{t}} + ⟨u_{t}^{1}, \frac{d}{dt}u_{t}^{2}⟩_{μ_{t}} = ⟨P_{μ_{t}}(\frac{d}{dt}u_{t}^{1}), u_{t}^{2}⟩_{μ_{t}} + ⟨u_{t}^{1}, P_{μ_{t}}(\frac{d}{dt}u_{t}^{2})⟩_{μ_{t}} = 0.

□

As a direct consequence we get the uniqueness of the parallel transport.

Corollary 6.24 (Uniqueness of parallel transport). Let μ_{t} be a regular curve and u ∈ Tan_{μ_{0}}(𝒫_{2}(ℝ^{d})). Then there exists at most one parallel transport t → u_{t} along μ_{t} such that u_{0} = u.

Proof. Suppose that u_{t}^{i}, i = 1, 2, are two parallel transports such that u_{0}^{i} = u, i = 1, 2. Then we know from the previous proposition that u_{t}^{1} − u_{t}^{2} is a parallel transport, too. Since we know also that the norm is preserved in time, we get ∥u_{t}^{1}−u_{t}^{2}∥_{μ_{t}} = ∥u−u∥_{μ_{0}} = 0. □

Observe that for parallel transports we have an explicit bound on the norm of \frac{d}{dt}u_{t}, which depends only on the Lipschitz constant of the vectors v_{t}.

Proposition 6.25. Let t → μ_{t} be a regular curve and t → u_{t} a parallel transport along it. Then it holds

(6.24)  ||\frac{d}{dt}u_{t}||_{μ_{t}} ≤ ∥u_{0}∥_{μ_{0}} Lip(v_{t}), a.e. t ∈ [0, 1].

Proof. We will prove that equation (6.24) is fulfilled at any Lebesgue point t of the function s → Lip(v_{s}). Fix such t and observe that it holds

∥u_{s} ∘ T(t, s, ·) − u_{t}∥_{μ_{t}} ≤ ∥P_{μ_{t}}(u_{s} ∘ T(t, s, ·)) − u_{t}∥_{μ_{t}} + ∥P_{μ_{t}}^{⊥}(u_{s} ∘ T(t, s, ·))∥_{μ_{t}}.

Dividing by |s − t| and letting s → t we have that the first term goes to 0 by definition of parallel transport, while for the second one we have the following estimate, based on proposition 6.6:

lim_{s→t^{+}} \frac{∥P_{μ_{t}}^{⊥}(u_{s} ∘ T(t, s, ·))∥_{μ_{t}}}{s − t} ≤ lim_{s→t^{+}} \frac{1}{s − t}∥u_{0}∥_{μ_{0}} Lip(T(t, s, ·) − Id)
≤ lim_{s→t^{+}} ∥u_{0}∥_{μ_{0}} \frac{exp (∫_{t}^{s} Lip(v_{r})dr) − 1}{s − t} = ∥u_{0}∥_{μ_{0}} Lip(v_{t}).

The case s → t^{−} is analogous. □

A simple consequence of theorem 6.18 is the following stability result of the parallel transport.

Theorem 6.26 (Stability of parallel transport). Let (μ_{t}^{n}) be a sequence of regular curves such that for every t the sequence (μ_{t}^{n}) converges to some μ_{t} w.r.t. the weak topology of 𝒫_{2}(ℝ^{d}). Assume also that the limit curve μ_{t} is absolutely continuous and that it holds:

\begin{aligned}\overline{lim}_{n→∞} ∥v_{t}^{n}∥_{μ_{t}^{n}} & ≤ ∥v_{t}∥_{μ_{t}}, a.e. t ∈ [0, 1], \\ Lip(v_{t}^{n}) & ≤ g(t), ∀n ∈ ℕ, a.e. t ∈ [0, 1], \end{aligned}

where v_{t}^{n}, v_{t} are the velocity vectors of μ_{t}^{n}, μ_{t} respectively and g ∈ L^{1}(0, 1). Let u_{t}^{n} ∈ Tan_{μ_{t}^{n}}(𝒫_{2}(ℝ^{d})) be a sequence of parallel transports along μ_{t}^{n} and assume that

i) for every t ∈ [0, 1], every weak limit point of (u_{t}^{n}) is tangent, ii) for some t_{0} ∈ [0, 1] (u_{t_{0}}^{n}) weakly (resp. strongly) converges to some u_{t_{0}} ∈

Tan_{μ_{t_{0}}} (𝒫_{2}(ℝ^{d})). Then for every t ∈ [0, 1] (u_{t}^{n}) weakly (resp. strongly) converges to some u_{t} and the limit vector field is tangent.

Proof. We know from theorem 6.18 that the limit curve is regular and that the maps T^{n}(t, s, ·) converge strongly to the map T(t, s, ·) for any t, s ∈ [0, 1].

Possibly extracting a subsequence, not relabeled, we may assume that (u_{t}^{n}) weakly converges to some vector u_{t} ∈ L_{μ_{t}}^{2} for every t ∈ [0, 1] ∩ ℚ. In this hypothesis, the maps u_{s}^{n} ∘ T^{n}(t, s, ·) converge weakly to the map u_{s} ∘ T(t, s, ·) for any s, t ∈ [0, 1] ∩ ℚ, indeed they are clearly equibounded and it holds

⟨u_{s}^{n}∘T^{n}(t, s, ·), φ⟩_{μ_{t}^{n}} = ⟨u_{s}^{n}, φ∘T^{n}(s, t, ·)⟩_{μ_{s}^{n}} → ⟨u_{s}, φ∘T(s, t, ·)⟩_{μ_{s}} = ⟨u_{s}∘T(t, s, ·), φ⟩_{μ_{t}},

having used proposition 2.23 to get the strong convergence of φ ∘ T^{n}(s, t, ·) to φ ∘ T(s, t, ·).

Therefore from proposition 6.25 we get

∥u_{s} ∘T(t, s, ·)−u_{t}∥_{μ_{t}} ≤ \underline{lim}_{n→∞} ∥u_{s}^{n} ∘T^{n}(t, s, ·)−u_{t}^{n}∥_{μ_{t}^{n}} ≤ \underline{lim}_{n→∞} ∫_{t}^{s} Lip(v_{r}^{n})dr ≤ ∫_{t}^{s} g(r)dr,

so the vector field t → u_{t} is absolutely continuous.

Now observe that since t → u_{t}^{n} is a parallel transport along μ_{t}^{n}, from equation (6.23) we know that it holds:

⟨u_{s}^{n}, ∇η⟩_{μ_{s}^{n}} − ⟨u_{t}^{n}, ∇η⟩_{μ_{t}^{n}} = ∫_{t}^{s} ⟨u_{r}^{n}, ∇^{2}η · v_{r}^{n}⟩_{μ_{r}^{n}}dr, ∀η ∈ C_{c}^{∞}(ℝ^{d}), 0 ≤ t ≤ s ≤ 1.

The strong convergence of (v_{r}^{n}) to v_{r} implies the strong convergence of ∇^{2}η · v_{r}^{n} to ∇^{2}η · v_{r}, furthermore the \overline{lim} of the functions r → ⟨u_{r}^{n}, ∇^{2}η · v_{r}^{n}⟩_{μ_{r}^{n}} is dominated by the L^{1}(0, 1) function \overline{lim}_{n} ∥u_{r}^{n}∥_{μ_{r}^{n}} Lip(∇η)∥v_{r}^{n}∥_{μ_{r}^{n}} ≤ sup_{n} (∥u_{t_{0}}^{n}∥_{μ_{t_{0}}^{n}} ) Lip(∇η)∥v_{r}∥_{μ_{r}}, so by the dominated convergence theorem we can pass to the limit in the above equation and get that the limit vector field is a parallel transport.

Given that we know that (u_{t_{0}}^{n}) weakly converges to u_{t_{0}} and that the parallel transport is unique, we get that this result is independent on the subsequence chosen, so for every t ∈ [0, 1] the whole sequence (u_{t}^{n}) weakly converges to u_{t}. If the convergence of (u_{t_{0}}^{n}) to u_{t_{0}} is strong, from the fact that the parallel transport preserves the norm we get lim_{n} ∥u_{t}^{n}∥_{μ_{t}^{n}} = lim_{n} ∥u_{t_{0}}^{n}∥_{μ_{t}^{n}} = ∥u_{t_{0}}∥_{μ_{t_{0}}} = ∥u_{t}∥_{μ_{t}} which gives the strong convergence of (u_{t}^{n}) to u_{t}. □

It is important to underline that hypothesis (i) is strictly needed, because it is not possible to deduce that the limit vector field is tangent assuming only that u_{t_{0}} is tangent for some t_{0} ∈ [0, 1], not even if we assume strong convergence of (u_{t}^{n}) for every t ∈ [0, 1]. The following is an explicit counterexample.

Example 6.27 (The limit vector field is not always tangent). Observe that for a regular curve t → μ_{t} and a given u_{0} ∈ Tan_{μ_{0}}(𝒫_{2}(ℝ^{d})), the vector u_{0} ∘ T(1, 0, ·) belongs to L_{μ_{1}}^{2} but is, in general, not tangent. So choose such a μ_{t} and u_{0} and apply proposition 6.21 to approximate the curve with a sequence (μ_{t}^{n}) of regular curves with range on measures with finite support such that:

lim_{n→∞} sup_{t∈[0,1]} W (μ_{t}^{n}, μ_{t}) = 0,

supp(μ_{t}^{n}) ⊂ supp(μ_{t}), ∀t ∈ [0, 1],

\frac{d}{dt}μ_{t}^{n} + ∇ · (v_{t}μ_{t}^{n}) = 0,

where the v_{t} are the velocity vectors of t → μ_{t}.

It is clear that the (restriction of) the flow maps T(t, s, ·) of μ_{t} to supp(μ_{t}^{n}) defines the flow maps of μ_{t}^{n}.

Now let γ^{n} ∈ Opt(μ_{0}, μ_{0}^{n}) and define u_{0}^{n} := u_{0} ∘ (γ^{n})^{−1}: it is easy to check that (u_{0}^{n}) strongly converges to u_{0}. Now let u_{t}^{n} := u_{0}^{n} ∘ T(t, 0, ·) and observe that since supp μ_{t}^{n} is finite, it holds u_{t}^{n} ∈ Tan_{μ_{t}^{n}}(𝒫_{2}(ℝ^{d})), furthermore it is clear from the definitions that t → u_{t}^{n} is a parallel transport for any n ∈ ℕ.

It follows from the strong convergence of (T(t, 0, ·), μ_{t}^{n}) to (T(t, 0, ·), μ_{t}) that (u_{0}^{n} ∘ T(t, 0, ·), μ_{t}^{n}) strongly converge to (u_{0} ∘ T(t, 0, ·), μ_{t}) as n → ∞.

Given that we assumed u_{0} ∘ T(1, 0, ·) ∉ Tan_{μ_{1}}(𝒫_{2}(ℝ^{d})) we have that the sequence of parallel transports t → u_{t}^{n} converges strongly for any t, but the limit vector field is tangent at t = 0 but not at t = 1.

In view if this example, it is interesting to know under which additional conditions the limit vector field is tangent: proposition 6.13 tells us that the two following hypothesis are both sufficient:

• For a.e. t and every n there exists a transport map T_{t}^{n} from μ_{t} to μ_{t}^{n} such that the sequence (Lip(T_{t}^{n} − Id)) converges to 0 as n → ∞,

• For a.e. t and every n there exists a regular mollifiers (ρ_{t}^{n}) (regular in the sense that the induced measure is in 𝒫_{2}^{r}(ℝ^{d})) such that μ_{t}^{n} = μ_{t} ∗ ρ_{t}^{n} and W (ρ_{t}^{n}, δ_{0}) → 0 as n → ∞ for any t.

Now we turn to the proof of the existence of the parallel transport: t → μ_{t} will be a fixed regular curve, v_{t} its velocity vector field and T(t, s, x) its flow.

In order to enlighten the notation we define

\begin{aligned}D(t, s) & := exp (∫_{t}^{s} Lip(v_{r})dr) − 1, 0 ≤ t ≤ s ≤ 1, \\ D(t, s) & := D(s, t) 0 ≤ s ≤ t ≤ 1. \end{aligned}

Then we let τ_{t}^{s} be the “translation” from L_{μ_{t}}^{2} to L_{μ_{s}}^{2} given by T(t, s, ·):

τ_{t}^{s}(f) := τ_{T(t,s,·)}(f) = f ∘ T(t, s, ·).

Note that from the group property of T(t, s, ·) it follows

(6.25)  τ_{t}^{r} = τ_{s}^{r} ∘ τ_{t}^{s}, ∀t, s, r ∈ [0, 1].

Moreover we define

𝒫_{t}^{s}(u) := P_{μ_{s}}(τ_{t}^{s}(u)).

Observe that the maps 𝒫_{t}^{s} are non-expansive and that by inequality (6.14) and proposition 6.6 we get:

(6.26a)  ∥𝒫_{s}^{t}(w)∥_{μ_{t}} ≤ ∥w∥_{μ_{s}}D(t, s), t, s ∈ [0, 1], w ∈ Tan_{μ_{s}}^{⊥}(𝒫_{2}(ℝ^{d})),
(6.26b)  ∥τ_{t}^{s}(u) − 𝒫_{t}^{s}(u)∥_{μ_{s}} ≤ ∥u∥_{μ_{t}}D(t, s), t, s ∈ [0, 1], u ∈ Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})).

To prove the existence of the transport we proceed as in the first section: let 𝔓 be the direct sets of all the partitions of [0, 1], where Q ≥ P, P, Q ∈ 𝔓, if Q is a refinement of P, and, for P = {0 = t_{0} ≤ t_{1} ≤ ⋯ ≤ t_{N} = 1} ∈ 𝔓 and u ∈ Tan_{μ_{0}}(𝒫_{2}(ℝ^{d})) define P(u) ∈ Tan_{μ_{1}}(𝒫_{2}(ℝ^{d})) as:

P(u) := 𝒫_{t_{N−1}}^{1}(𝒫_{t_{N−2}}^{t_{N−1}}(⋯ 𝒫_{0}^{t_{1}}(u))).

Finally, let D^{2}(P) := ∑_{i} D^{2}(t_{i}, t_{i+1}).

We will prove first that there exists a unique limit T_{0}^{1}(u) ∈ Tan_{μ_{1}}(𝒫_{2}(ℝ^{d})) of P(u) as P varies in 𝔓, then we will define a curve t → u_{t} = T_{0}^{t}(u) ∈ Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})) by considering the restriction of t → μ_{t} to [0, t], and finally prove that this curve is the parallel transport of u along the curve t → μ_{t}.

Lemma 6.28. It holds

(6.27a)  D(t_{1}, s_{1}) ≤ D(t_{2}, s_{2}), ∀[t_{1}, s_{1}] ⊂ [t_{2}, s_{2}] ⊂ [0, 1],
(6.27b)  ∑_{i=1}^{n−1} D(t_{i}, t_{t+1}) ≤ D(t, s), t = t_{1} ≤ … ≤ t_{n} = s,
(6.27c)  lim_{P∈𝔓}D^{2}(P) = 0,
(6.27d)  lim_{s→t} \frac{D^{2}(t, s)}{|s − t|} = 0, a.e. t ∈ [0, 1].

Proof. Equation (6.27a) is clear. For (6.27b) we need to prove that e^{a} − 1 + e^{b} − 1 ≤ e^{a+b} − 1 for positive a, b, which is obvious.

The convexity of r → e^{r} − 1 in [0, ∫_{0}^{1} Lip(v_{t})dt] gives
(6.28)  D(t, s) ≤ (\frac{∫_{0}^{1} Lip(v_{r})dr − 1}{∫_{0}^{1} Lip(v_{r})dr}) ∫_{t}^{s} Lip(v_{r})dr,

from which, taking the integrability of Lip(v_{t}) into account, (6.27d) follows at every Lebesgue point of t ↦ Lip(v_{t}). Finally, from (6.28) we get

∑_{i=0}^{N−1} D^{2}(t_{i+1}, t_{i}) ≤ C ∑_{i=0}^{N−1}(∫_{t_{i}}^{t_{i+1}} Lip(v_{r})dr)^{2} ≤ C max_{i} {∫_{t_{i}}^{t_{i+1}} Lip(v_{r}) dr} ∫_{0}^{1} Lip(v_{r}) dr,

from which (6.27c) follows, taking the absolute continuity property of the integral into account. □

Lemma 6.29. Let 0 ≤ s_{1} ≤ s_{2} ≤ s_{3} ≤ 1 and let u ∈ Tan_{μ_{s_{1}}} (𝒫_{2}(ℝ^{d})). Then it holds:

(6.29)  ||𝒫_{s_{1}}^{s_{3}}(u) − 𝒫_{s_{2}}^{s_{3}}(𝒫_{s_{1}}^{s_{2}}(u))||_{μ_{s_{3}}} ≤ ∥u∥_{μ_{s_{1}}} D(s_{1}, s_{2})D(s_{2}, s_{3}).

Proof. Observe that

𝒫_{s_{1}}^{s_{3}}(u) − 𝒫_{s_{2}}^{s_{3}}(𝒫_{s_{1}}^{s_{2}}(u)) = 𝒫_{s_{2}}^{s_{3}}(τ_{s_{1}}^{s_{2}}(u) − 𝒫_{s_{1}}^{s_{2}}(u)),

and that τ_{s_{1}}^{s_{2}}(u) − 𝒫_{s_{1}}^{s_{2}}(u) ∈ Tan_{μ_{s_{2}}}^{⊥} (𝒫_{2}(ℝ^{d})). Therefore the thesis follows by a direct application of inequalities (6.26). □

Corollary 6.30. Let P = {t = t_{0} ≤ ⋯ ≤ t_{n} = s} be a partition of [t, s] ⊂ [0, 1] and let Q be a refinement of P. Then it holds

(6.30)  ∥P(u) − Q(u)∥_{μ_{1}} ≤ ∥u∥_{μ_{0}}D^{2}(P),

for every u ∈ Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})).

Proof. Without loss of generality we may assume [t, s] = [0, 1]. Fix i < n such that (t_{i}, t_{i+1}) contains some element of Q and write Q ∩ [t_{i}, t_{i+1}] = {t_{i} = s_{i,0} < s_{i,1} < ⋯ < s_{i,k(i)} = t_{i+1}} for some k(i) ≥ 1. Now, we claim that

(6.31) ∥𝒫_{s_{i,0}}^{s_{i,k(i)}}(u_{t_{i}}) − 𝒫_{s_{i,k_{i}−1}}^{s_{i,k_{i}}}(𝒫_{s_{i,k_{i}−2}}^{s_{i,k_{i}−1}}(⋯ (𝒫_{s_{i,0}}^{s_{i,1}}(u_{t_{i}}))))∥_{μ_{t_{i+1}}} ≤ ∥u_{t_{i}}∥_{μ_{t_{i}}} D^{2}(t_{i}, t_{i+1})

for all u_{t_{i}} ∈ Tan_{μ_{t_{i}}} (𝒫_{2}(ℝ^{d})). Indeed, the left hand side of (6.31) can be estimated by

∥𝒫_{s_{i,0}}^{s_{i,k(i)}}(u_{t_{i}}) − 𝒫_{s_{i,k(i)−1}}^{s_{i_{k(i)}}}(𝒫_{s_{i,0}}^{s_{i,k(i)−1}}(u_{t_{i}}))∥_{μ_{t_{i+1}}}
+ ∥𝒫_{s_{i,k(i)−1}}^{s_{i,k(i)}}(𝒫_{s_{i,0}}^{s_{i,k(i)−1}}(u_{t_{i}})) − 𝒫_{s_{i,k(i)−1}}^{s_{i,k(i)}}(𝒫_{s_{i,k(i)−2}}^{s_{i,k(i)−1}}(⋯ (𝒫_{s_{i,0}}^{s_{i,1}}(u_{t_{i}}))))∥_{μ_{t_{i+1}}}
≤ ∥u_{t_{i}}∥_{μ_{t_{i}}} D(s_{i,0}, s_{i,k(i)−1})D(s_{i,k(i)−1}, s_{i,k(i)})
+ ∥𝒫_{s_{i,0}}^{s_{i,k(i)−1}}(u_{t_{i}}) − 𝒫_{s_{i,k(i)−2}}^{s_{i,k(i)−1}}(𝒫_{s_{i,k(i)−3}}^{s_{i,k(i)−2}}(⋯ (𝒫_{t_{i}}^{s_{i,0}}(u_{t_{i}}))))∥_{μ_{t_{i+1}}}
≤ ⋯
≤ ∥u_{t_{i}}∥_{μ_{t_{i}}} ∑_{j=0}^{k(i)−1} D(s_{i,0}, s_{i,j})D(s_{i,j}, s_{i,j+1}) ≤ ∥u_{t_{i}}∥_{μ_{t_{i}}} D(t_{i}, t_{i+1}) ∑_{j=0}^{k(i)−1} D(s_{i,j}, s_{i,j+1})
≤ ∥u_{t_{i}}∥_{μ_{t_{i}}} D^{2}(t_{i}, t_{i+1}).

Now, let us assume that (t_{0}, t_{1}) contains some element of Q and let P′ = [t_{1}, 1] ∩ P, Q′ = [t_{1}, 1] ∩ Q, u ∈ Tan_{μ_{0}}(𝒫_{2}(ℝ^{d})) and v, w ∈ Tan_{μ_{t_{1}}} (𝒫_{2}(ℝ^{d})) be such that P(u) = P′(v) and Q(u) = Q′(w). Then, the inequality (6.31) with i = 0 reads

∥v − w∥_{μ_{t_{1}}} ≤ ∥u∥_{t_{0}}D^{2}(t_{0}, t_{1}),

(the estimate is trivial if Q′ ∩ (t_{0}, t_{1}) = ∅, because v = w) so that

∥P(u) − Q(u)|_{μ_{tn}} ≤ ∥P′(v) − Q′(v)∥_{μ_{tn}} + ∥Q′(v) − Q′(w)∥_{μ_{tn}}
≤ ∥P′(v) − Q′(v)∥_{μ_{tn}} + ∥u∥_{t_{0}}D^{2}(t_{0}, t_{1}).

Since ∥v∥_{t_{1}} ≤ ∥u∥_{t_{0}} we can apply repeatedly (6.31) in the intervals (t_{i}, t_{i+1}) to obtain ∥P(u) − Q(u)∥_{μ_{1}} ≤ ∥u∥_{μ_{0}}D^{2}(P). □

Theorem 6.31 (Existence of the limit of P(u_{0})). Let t → μ_{t} be a regular curve and u_{0} ∈ Tan_{μ_{0}}(𝒫_{2}(ℝ^{d})). Then there exists the limit of P(u_{0}) as P varies in the direct set 𝔓.

Proof. It follows directly from the previous corollary and from lemma 6.28. □

Define T_{0}^{1}(u_{0}) as the vector obtained by the limit process described above, and observe that by repeating the arguments to the restriction of t → μ_{t} to the interval [t, s], we can define a map T_{t}^{s} : Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})) → Tan_{μ_{s}}(𝒫_{2}(ℝ^{d})). Furthermore, by considering the curve t → μ_{1−t}, we can define the maps T_{t}^{s} even for t > s.

Proposition 6.32 (Group property). Let t → μ_{t} be a regular curve and T_{t}^{s} : Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})) → Tan_{μ_{s}}(𝒫_{2}(ℝ^{d})) be defined as above. Then

(6.32)  T_{t}^{s} ∘ T_{r}^{t} = T_{r}^{s}, ∀r, s, t ∈ [0, 1].

Proof. Let us first assume r ≤ t ≤ s. In this case it is sufficient to observe that, by definition of limit over a direct set, to take the limit over all the partitions of [r, s] is equal to take the limit over the set of partitions which contain the point t. The thesis then follows easily.

For the general case it is sufficient to prove that T_{t}^{s} = (T_{s}^{t})^{−1}, or, without loss of generality, that T_{0}^{1} = (T_{1}^{0})^{−1}. The latter equation will follow if we show that

(6.33)  lim_{P∈𝔓}∥u − P^{−1}(P(u))∥_{μ_{0}} = 0, ∀u ∈ Tan_{μ_{0}}(𝒫_{2}(ℝ^{d})),

where P^{−1} : Tan_{μ_{1}}(𝒫_{2}(ℝ^{d})) → Tan_{μ_{0}}(𝒫_{2}(ℝ^{d})) is defined as

P^{−1}(u) := 𝒫_{t_{1}}^{0}(𝒫_{t_{2}}^{t_{1}}(⋯ 𝒫_{1}^{t_{n}−1}(u)))

for the partition P = {0 = t_{0} ≤ ⋯ t_{n} = 1} (and in particular is not the functional inverse of u → P(u)).

Observe that for any u ∈ Tan_{μ_{t_{i}}} (𝒫_{2}(ℝ^{d})) the identities u = 𝒫_{t_{i+1}}^{t_{i}}(τ_{t_{i}}^{t_{i+1}}(u)) and 𝒫_{t_{i}}^{t_{i+1}}(u) − τ_{t_{i}}^{t_{i+1}}(u) ∈ Tan_{μ_{t_{i+1}}}^{⊥} (𝒫_{2}(ℝ^{d})), in conjunction with inequalities (6.26), yield

\begin{matrix}∥𝒫_{t_{i+1}}^{t_{i}}(𝒫_{t_{i}}^{t_{i+1}}(u)) − u∥_{μ_{t_{i}}} = & ∥𝒫_{t_{i+1}}^{t_{i}}(𝒫_{t_{i}}^{t_{i+1}}(u) − τ_{t_{i}}^{t_{i+1}}(u))∥_{μ_{t_{i}}} \\ ≤ & ∥𝒫_{t_{i}}^{t_{i+1}}(u) − τ_{t_{i}}^{t_{i+1}}(u)∥_{μ_{t_{i}}} D(t_{i}, t_{i+1}) \\ ≤ & ∥u∥_{μ_{t_{i}}} D^{2}(t_{i}, t_{i+1}). \end{matrix}

For any u ∈ Tan_{μ_{0}}(𝒫_{2}(ℝ^{d})) we obtain

∥u − 𝒫_{t_{1}}^{0}(⋯ ( 𝒫_{1}^{t_{n−1}}(P(u)))∥_{μ_{0}}
≤ ∥u − 𝒫_{t_{1}}^{0}(𝒫_{0}^{t_{1}}(u))∥_{μ_{0}} + ∥𝒫_{t_{1}}^{0}(𝒫_{0}^{t_{1}}(u)) − 𝒫_{t_{1}}^{0}(⋯ (𝒫_{1}^{t_{n−1}}(P(u))))∥_{μ_{0}}
≤ ∥u∥_{μ_{0}}D^{2}(0, t_{1}) + ∥v − 𝒫_{t_{2}}^{t_{1}}(⋯ (𝒫_{t_{n−1}}^{1}(P′(v))))∥_{μ_{t_{1}}} ,

where v = 𝒫_{0}^{t_{1}}(u) and P′ = {t_{1} < ⋯ < t_{n}} (so that P′(v) = P(u)). Since ∥v∥_{μ_{t_{1}}} ≤ ∥u∥_{μ_{0}} we can continue in this way, to arrive at

∥u − 𝒫_{t_{1}}^{0}(⋯ (𝒫_{1}^{t_{n−1}}(P(u))))∥_{μ_{0}} ≤ ∥u∥_{μ_{0}}D^{2}(P)

and this, taking (6.27c) into account, leads to (6.33). □

Proposition 6.33 (The limit process produces the parallel transport). Let t → μ_{t} be a regular curve, u_{0} ∈ Tan_{μ_{0}}(𝒫_{2}(ℝ^{d})) and let the maps T_{t}^{s} be defined as above. Then the vector field t → u_{t} := T_{0}^{t}(u_{0}) is the parallel transport of u_{0} along the curve.

Proof. Consider any interval [t, s] ⊂ [0, 1], its trivial partition P = {t, s} and any (finer) partition Q. Applying inequality (6.30) and passing to the limit on Q we get

(6.34)  ∥𝒫_{t}^{s}(u) − T_{t}^{s}(u)∥_{μ_{s}} ≤ ∥u∥_{μ_{t}}D^{2}(t, s), ∀u ∈ Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})).

Coupling this equation with inequality (6.26b) we get

\begin{matrix}∥τ_{t}^{s}(u) − T_{t}^{s}(u)∥_{μ_{s}} ≤ & ∥τ_{t}^{s}(u) − 𝒫_{t}^{s}(u)∥_{μ_{s}} + ∥𝒫_{t}^{s}(u) − T_{t}^{s}(u)∥_{μ_{s}} \\ ≤ & ∥u∥_{μ_{t}}D(t, s) (1 + D(0, 1)) , ∀u ∈ Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})), \end{matrix}

from which it easily follows the absolute continuity of t → T_{0}^{t}(u_{0}).

Now pick a Lebesgue point t of the function t → Lip(v_{t}) and observe that inequality (6.34) gives

lim_{s→t} \frac{∥𝒫_{t}^{s}(u) − T_{t}^{s}(u)∥_{μ_{s}}}{|s − t|} = 0,

for every u ∈ Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})), and so in particular for u = T_{0}^{t}(u), therefore to conclude it is sufficient to prove that

lim_{s→t} P_{μ_{t}} (\frac{τ_{s}^{t}(𝒫_{t}^{s}(u)) − u}{s − t} ) = 0, ∀u ∈ Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})).

Observe that 𝒫_{t}^{s}(u) − τ_{t}^{s}(u) ∈ Tan_{μ_{s}}^{⊥}(𝒫_{2}(ℝ^{d})), therefore from inequalities (6.26) we get

∥P_{μ_{t}}(τ_{s}^{t}(𝒫_{t}^{s}(u)) − u)∥_{μ_{t}} = ∥𝒫_{s}^{t}(𝒫_{t}^{s}(u) − τ_{t}^{s}(u))∥_{μ_{t}}

≤ ∥𝒫_{t}^{s}(u) − τ_{t}^{s}(u)∥_{μ_{t}}D(t, s) ≤ ∥u∥_{μ_{t}}D^{2}(t, s).

□

Remark 6.34 (Parallel transport along a flow). The above arguments works as well if, instead than assuming that the curve μ_{t} is regular, we assume the existence of a family of maps X(t, s, x) satisfying X(t, s, ·)_{#}μ_{t} = μ_{s}, having the group property X(t, s, X(r, t, x)) = X(r, s, x) and such that the Lipschitz constant of X(t, s, ·) − Id is bounded by a certain function D(t, s) having the properties (6.27) (i.e. we just drop the condition that the velocity field of X is the tangent velocity vector of μ_{t}). The result would be the existence of a vector field t → u_{t} ∈ Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})) such that t → u_{t} ∘ X(s, t, ·) ∈ L_{μ_{s}}^{2} is absolutely continuous for any s and satisfying the equation

P_{μ_{t}} (lim_{h→0} \frac{u_{t+h} ∘ X(t, t + h, ·) − u_{t}}{h} ) = 0,

for a.e. t. For such vector fields, it is possible to prove, with exactly the same arguments, uniqueness, linearity and conservation of scalar product.

We will show in the next section that our choice of the flows T related to the tangent vector field is more natural, because it induces the Levi-Civita connection on 𝒫_{2}(ℝ^{d}).

Before analyzing some examples of parallel transport, we want to point out that the construction we did of parallel tranport along a regular curve, allows a little generalization to the case of forward parallel transport along a locally regular curve. The question is the following. Consider an absolutely continuous curve μ_{t} on [0, T] such that the function t → Lip(v_{t}) belongs to L_{loc}^{1}((0, T]) (we will see in example 6.39 that this is the case for a geodesic on [0, 1] for any T ∈ (0, 1)). We say that [0, T] ∋ t → u_{t} ∈ Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})) is a parallel transport if it is a parallel transport on the interval (0, T] (which makes sense, due to the locality of the definition of parallel transport) and u_{t} strongly converge to u_{0} as t → 0. Having this definition in mind, two questions come out naturally: the first one is whether there exists the parallel transport along μ_{t} of a vector in Tan_{μ_{0}}(𝒫_{2}(ℝ^{d})), which we call forward parallel transport, the second one is whether there exists the parallel transport of a vector in Tan_{μ_{T}} (𝒫_{2}(ℝ^{d})), which we call backward parallel transport. We are going to prove here that the forward parallel transport always exists, while the same is not true for the backward one. This reflects what we already said at the end of section 2, where we proved that a sequence of tangent vectors doesn’t necessarily converge to a tangent vector (and thus the backward parallel transport may not exist), and that every tangent vector may be approximated by tangent vectors (and thus the forward parallel transport exists).

For the proof of existence of the forward parallel transport, we will need the following technical result which is of its own interest.

Lemma 6.35. Let t → μ_{t} be a regular curve and T_{t}^{s} the optimal transport maps along it. Then it holds

(6.35)  ∥T_{t}^{s}(∇φ) − ∇φ∥_{μ_{s}} ≤ Lip(∇φ)L_{t}^{s}(μ_{r}), ∀φ ∈ C_{c}^{∞}(ℝ^{d}),

where L_{t}^{s}(μ_{r}) is the length of μ_{r} restricted to the interval [t, s].

Proof. Observe that s → T_{t}^{s}(∇φ) − ∇φ ∈ Tan_{μ_{s}}(𝒫_{2}(ℝ^{d})) is an absolutely continuous vector field along μ_{t}. The conclusion follows from the differential inequality:

\frac{d}{ds}∥T_{t}^{s}(∇φ) − ∇φ∥_{μ_{s}}^{2} = 2⟨T_{t}^{s}(∇φ) − ∇φ, \frac{d}{ds}(T_{t}^{s}(∇φ) − ∇φ)⟩_{μ_{s}}

= 2⟨T_{t}^{s}(∇φ) − ∇φ, \frac{d}{ds}(T_{t}^{s}(∇φ)) − ∇^{2}φ · v_{s}⟩_{μ_{s}}

= 2⟨T_{t}^{s}(∇φ) − ∇φ, P_{μ_{s}}(\frac{d}{ds}(T_{t}^{s}(∇φ)) − ∇^{2}φ · v_{s})⟩_{μ_{s}}

= −2⟨T_{t}^{s}(∇φ) − ∇φ, P_{μ_{s}}(∇^{2}φ · v_{s})⟩_{μ_{t}}

= −2⟨T_{t}^{s}(∇φ) − ∇φ, ∇^{2}φ · v_{s}⟩_{μ_{t}}

≤ 2∥T_{t}^{s}(∇φ) − ∇φ∥_{μ_{s}} Lip(∇φ)∥v_{s}∥_{μ_{s}}.

□

Proposition 6.36 (Forward parallel transport). Let [0, T] ∋ t → μ_{t} be an absolutely continuous curve such that the function Lip(v_{t}) belongs to L_{loc}^{1}((0, T]) and let u_{0} ∈ Tan_{μ_{0}}(𝒫_{2}(ℝ^{d})). Then there exists a parallel transport (0, T) ∋ t → u_{t} ∈ Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})) such that u_{t} converges strongly to u_{0} as t → 0.

Proof. Start assuming that u_{0} is the gradient of φ ∈ C_{c}^{∞}(ℝ^{d}). Fix ε > 0, think ∇φ as a vector in Tan_{μ_{ε}}(𝒫_{2}(ℝ^{d})) and define the vectors u_{t}^{ε} := T_{ε}^{t}(∇φ) for any t ∈ [ε, T], so that we have u_{ε}^{ε} = ∇φ. From

∥u_{t}^{ε′} −u_{t}^{ε}∥_{μ_{t}} = ∥u_{ε}^{ε′} −u_{ε}^{ε}∥_{μ_{ε}} = ∥T_{ε′}^{ε}(∇φ)−∇φ∥_{μ_{ε}} ≤ Lip(∇φ)ω(ε) 0 < ε′ ≤ ε ≤ t ≤ T,

with ω(ε) := ∫_{0}^{ε} ∥v_{t}∥_{μ_{t}}dt, we get that for any t, the family {u_{t}^{ε}} converges in Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})), as ε → 0, to a vector u_{t} satisfying ∥u_{t}^{ε} − u_{t}∥_{μ_{t}} ≤ Lip(∇φ)ω(ε). The limit vector field u_{t} is easily seen to be a parallel transport in the interval (0, T]: indeed from the uniform bound (6.24) we get its local absolute continuity, and we conclude by the stability of the solutions of (6.23).

From

∥u_{t}∥_{μ_{t}} = lim_{ε} ∥u_{t}^{ε}∥_{μ_{t}} = lim_{ε} ∥u_{ε}^{ε}∥_{μ_{ε}} = lim_{ε} ∥∇φ∥_{μ_{ε}}

we get that the norm of u_{t} is constant, and equal to ∥∇φ∥_{μ_{0}}. Finally it holds

⟨u_{ε}, η⟩_{μ_{ε}} = ⟨u_{ε} − u_{ε}^{ε}, η⟩_{μ_{ε}} + ⟨u_{ε}^{ε}, η⟩_{μ_{ε}} = R_{ε} + ⟨∇φ, η⟩_{μ_{ε}} ∀η ∈ C_{c}^{∞}(ℝ^{d}, ℝ^{d}),

where the term R_{ε} is bounded by ∥u_{t} − u_{t}^{ε}∥_{μ_{t}} sup |η| ≤ ω(ε) Lip(∇φ) sup |η|.

For the general case, just approximate u_{0} with smooth gradients u_{0}^{n}, apply the construction above to obtain the existence of forward parallel transports t → u_{t}^{n} of u_{0}^{n} and use the fact that (clearly) ∥u_{t}^{n} − u_{t}^{m}∥_{μ_{t}} = ∥u_{0}^{n} − u_{0}^{m}∥_{μ_{0}} to get that for any t the sequence (u_{t}^{n}) strongly converges to some u_{t} such that ∥u_{t}∥_{μ_{t}} = ∥u_{0}∥_{μ_{0}}. By the stability argument used above we get that t → u_{t} is a parallel transport on (0, T], so we need just to prove that u_{t} weakly converges to u_{0} as t → 0. To prove this, observe that since [0, T] ∋ t → u_{t}^{n} is a forward parallel transport, passing to the limit as t → 0 in (6.23) we get that for every η ∈ C_{c}^{∞}(ℝ^{d}) it holds:

⟨u_{t}^{n}, ∇η⟩_{μ_{t}} − ⟨u_{0}^{n}, ∇η⟩_{μ_{0}} = ∫_{0}^{t} ⟨u_{r}^{n}, ∇^{2}η · v_{r}⟩_{μ_{r}}dr ≤ ∥u_{t}^{n}∥_{μ_{t}} Lip(∇η)ω(t).

Letting n → ∞ in the above inequality the weak convergence follows. □

Now we turn to the counterexample to the existence of the backward parallel transport. As we will see, it is possible that a parallel transport u_{t} exists for positive t, that the vectors u_{t} converge strongly to some vector u_{0} as t → 0, but such that the vector u_{0} is not a tangent vector: this shows that the general problem of existence of the parallel transport is, in general, intrisically prohibited by the geometry of 𝒫_{2}(ℝ^{d}). Observe that the curve considered is a geodesic.

Example 6.37. Let Q = [0, 1]×[0, 1] be the unit square in ℝ^{2} and let T_{i}, i = 1, 2, 3, 4, be the four open triangles in which Q is divided by its diagonals. Let μ_{0} := χ_{Q}ℒ^{2} and define the function v : Q → ℝ^{2} as the gradient of the convex map max{|x|, |y|}, as in the figure. Set also u = v^{⊥}, the rotation by π/2 of v, in Q and u = 0 out of Q. Notice that u is orthogonal to Tan_{μ}(𝒫_{2}(ℝ^{d})), since it holds ∇ · (uμ) = 0.

Set μ_{t} := (Id+tv)_{#}μ_{0} and observe that, for positive t, the support Q_{t} of μ_{t} is made of 4 connected components, each one the translation of one of the sets T_{i}, and that μ_{t} = χ_{Q_{t}}ℒ^{2}.

It is immediate to check that the velocity vectors of μ_{t} are given by v_{t} := v ∘ (Id + tv)^{−1}, so that Lip(v_{t}) = t^{−1} and μ_{t} is locally regular in (0, 1), and that the flow maps of μ_{t} in (0, 1] are given by

T(t, s, ·) = (Id + sv) ∘ (Id + tv)^{−1}, ∀t, s ∈ (0, 1].

Now, set u_{t} := u ∘ T(t, 0, ·) and notice that u_{t} is tangent at μ_{t}, because u_{t} is constant in the connected components of the support of μ_{t}. Since u_{t+h} ∘ T(t, t + h, ·) = u_{t}, we obtain that u_{t} is a parallel transport in (0, 1]. Furthermore, since u_{t} converges to u as t → 0 and u ∉ Tan_{μ}(𝒫_{2}(ℝ^{d})), there is no way to extend u_{t} to a continuous tangent vector field on the whole [0, 1].

In the rest of the section we analyze some examples of parallel transport.

Example 6.38 (Equation in the smooth case). Assume that u_{t}(x) = ∇φ(t, x) for a certain function φ ∈ C^{2}([0, 1] × ℝ^{d}, ℝ) with uniformly bounded second derivatives. Then it is easy to see that equation (6.21) becomes:

(6.36)  ∇ · (μ_{t}(∂_{t}∇φ + ∇_{x}^{2}φ · v_{t})) = 0.

or, which is the same

(6.37)  ∫_{0}^{1} ∫ ⟨∂_{t}∇φ + ∇_{x}^{2}φ · v_{t}, ∇η⟩dμ_{t}dt = 0, ∀η ∈ C_{c}^{∞}((0, 1) × ℝ^{d}, ℝ)

Observe that these equations are valid without any regularity assumption on the measures involved.

Example 6.39 (Geodesics). Consider a geodesic t → μ_{t} defined on the interval [0, 1]: we want to prove that in any interval of the form [ε, 1 − ε] it is regular and its velocity vectors are a parallel transport on each interval of this kind.

Fix t ∈ (0, 1); we know from proposition 2.11 that there exists only one optimal plan between μ_{t} and μ_{1} and that this plan is induced by a Lipschitz map T_{t} with Lipschitz constant bounded by t^{−1}. We know also that for s ∈ [t, 1] it holds μ_{s} = (Id + \frac{s−t}{1−t}(T_{t} − Id))_{#}μ_{t}, the transport map being optimal. Calculating the velocity vector v_{t} as limit of the optimal transport maps, we get v_{t} = (1 − t)^{−1}(T_{t} − Id), therefore its Lipschitz constant is bounded by (1 + t)(t(1 − t))^{−1} (actually, since T_{t} is monotone, it can be proved that Lip(v_{t}) ≤ (t(1 − t))^{−1}). Our claim on the regularity of μ_{t} in [ε, 1 − ε] follows.

Now assume that the geodesic [0, 1] ∋ t → μ_{t} is induced by a Lipschitz optimal map T. In this case its flow is given by

T(t, s, ·) = (Id + s(T − Id)) ∘ (Id + t(T − Id))^{−1},

and the velocity vectors satisfy

v_{s} = v_{t} ∘ T (t, s, ·),

therefore a direct calculation shows that v_{t} is a parallel transport.

Example 6.40 (Constant vector fields). Let t → μ_{t} be a regular curve, v ∈ ℝ^{d} and C_{v} the function on ℝ^{d} constantly equal to v. Define u_{0} := C_{v} ∈ Tan_{μ_{0}}(𝒫_{2}(ℝ^{d})). We claim that the parallel transport u_{t} of u_{0} along μ_{t} is given by u_{t} = C_{v}, for all t ∈ [0, 1].

The proof is immediate: it is sufficient to observe that u_{t}(x) = ∇φ(t, x), where φ(t, x) = ⟨x, v⟩, and to verify that φ satisfies equation (6.36).

Example 6.41 (Translations). Choose μ_{0} ∈ 𝒫_{2}(ℝ^{d}) and an absolutely continuous curve [0, 1] ∋ t → γ(t) ∈ ℝ^{d} such that γ(0) = 0. As before, for any vector v ∈ ℝ^{d}, let C_{v} : ℝ^{d} → ℝ^{d} be the function constantly equal to v. Define μ_{t} := (Id + C_{γ(t)})_{#}μ_{0}, so that μ_{t} is the translation of μ_{0} by the vector γ(t).

It is clear that in this case the curve μ_{t} is regular and that its flow maps are given by

(6.38)  T(t, s, x) = x + γ(s) − γ(t).

Furthermore, it is easy to check that the composition with T(t, s, x) defines an isometry from Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})) to Tan_{μ_{s}}(𝒫_{2}(ℝ^{d})), therefore (by construction) the parallel transport maps coincide with the composition with the flow maps: i.e. for a given u_{0} ∈ Tan_{μ_{0}}(𝒫_{2}(ℝ^{d})), the parallel tranport u_{t} of u_{0} along μ_{t} is given by

(6.39)  u_{t}(x) = u_{0}(x − γ(t)).

Example 6.42 (Separate sets). The parallel transport behaves independently on different connected components of supp μ_{t}. Indeed, recall remark 6.17 and observe that, with the same notation, a simple cut-off argument shows that u ∈ Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})) if and only if u ∈ Tan_{μ_{t}^{i}}(𝒫_{2}(ℝ^{d})) for any i = 1, …, n. Therefore a straightforward application of the definition of parallel transport gives that t → u_{t} ∈ Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})) is a parallel transport along μ_{t} if and only if t → u_{t} ∈ Tan_{μ_{t}^{i}}(𝒫_{2}(ℝ^{d})) is a parallel transport along μ_{t}^{i} for any i = 1, …, n.

5. Covariant derivative and curvature operator

In this section we analyze the covariant derivative and the curvature tensor induced by the parallel transport we defined.

It is well known that, in the classical Riemannian setting, the definition of parallel transport leads to the one of covariant derivative via the formula

(6.40)  ∇_{γ̇(t)}u(t) := lim_{s→t} \frac{T_{s}^{t}(u(s)) − u(t)}{s − t},

where γ(t) is a smooth curve, u(t) ∈ T_{γ(t)}M is a smooth vector field and T_{s}^{t}, for any s, t, is the parallel transport map from T_{γ(s)}M to T_{γ(t)}M along γ.

The same construction may be used in the Wasserstein setting:

Definition 6.43 (Covariant derivative). Let μ_{t} be a regular curve, let v_{t} ∈ Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})) be its velocity vector and let u_{t} ∈ Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})) be an absolutely continuous vector field along μ_{t}. The covariant derivative of u_{t} along μ_{t} is:

∇_{v_{t}}u_{t} := lim_{s→t} \frac{T_{s}^{t}(u_{s}) − u_{t}}{s − t} ,

where T_{s}^{t} are the parallel transport maps along μ_{t} and the derivative takes place in L_{μ_{t}}^{2}.

Using the definition of absolutely continuous vector field, it is not difficult to check that the covariant derivative exists for a.e. t and that the function t ↦ ∥∇_{v_{t}}u_{t}∥_{μ_{t}} is integrable. Indeed, inequality (6.34) implies that the covariant derivative satisfies:

(6.41)  ∇_{v_{t}}u_{t} = lim_{s→t} \frac{𝒫_{s}^{t}(u_{s}) − u_{t}}{s − t} = P_{μ_{t}} (lim_{s→t} \frac{u_{s} ∘ T(t, s, ·) − u_{t}}{s − t} ) .

If the vector field u_{t} is given by the gradient of smooth functions, i.e. if u_{t} = ∇_{x}φ_{t}(x) for some φ_{t} ∈ C_{c}^{∞}(ℝ^{d}) smoothly varying in time, the previous equation reads as

(6.42)  ∇_{v_{t}}u_{t} = P_{μ_{t}}(∂_{t}∇φ_{t} + ∇_{x}^{2}φ_{t} · v_{t}).

Equation (6.42) and the analogous one (6.36) were first given in [43], although from a formal viewpoint and under stronger assumptions on the measures μ_{t}.

Having defined the covariant derivative, our first goal is to prove that it is the Levi-Civita connection on (𝒫_{2}(ℝ^{d}), W_{2}). Recalling the discussion made for the classical case of Riemanniann manifolds, we need to prove that it is compatible with the metric and torsion-free. The compatibility with the metric is a simple consequence of the definition: indeed, for a given couple of absolutely continuous vector fields u_{t}^{1}, u_{t}^{2} ∈ Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})) along the regular curve μ_{t}, we have:

\begin{aligned}\frac{d}{dt}⟨u_{t}^{1}, u_{t}^{2}⟩_{μ_{t}} & = ⟨\frac{d}{dt}u_{t}^{1}, u_{t}^{2}⟩_{μ_{t}} + ⟨u_{t}^{1}, \frac{d}{dt}u_{t}^{2}⟩_{μ_{t}} \\ & = ⟨P_{μ_{t}} (\frac{d}{dt}u_{t}^{1}) , u_{t}^{2}⟩_{μ_{t}} + ⟨u_{t}^{1}, P_{μ_{t}} (\frac{d}{dt}u_{t}^{2}) ⟩_{μ_{t}} \\ & = ⟨∇_{v_{t}}u_{t}^{1}, u_{t}^{2}⟩_{μ_{t}} + ⟨u_{t}^{1}, ∇_{v_{t}}u_{t}^{2}⟩_{μ_{t}}, \end{aligned} ; (6.43)

having used the Leibnitz rule (6.15) and the fact that both vector fields are tangent.

To prove the torsion-free identity, we need first to understand how to calculate the Lie bracket of two vector fields. To this aim, let μ_{t}^{i}, i = 1, 2, be two regular curves such that μ_{0}^{1} = μ_{0}^{2} =: μ and let u_{t}^{i} ∈ Tan_{μ_{t}^{i}}(𝒫_{2}(ℝ^{d})) be two absolutely continuous vector fields along μ_{t}^{i} satisfying u_{0}^{i} = v_{0}^{1−i}, where v_{t}^{i} are the tangent velocity fields of μ_{t}^{i}. We assume that both the velocity fields v_{t}^{i} of μ_{t}^{i} and the derivative of u_{t}^{i} exists for all t and are continuous in time w.r.t. the strong convergence of maps, so that the Leibnitz rule (6.15) holds for all the times (and not just as functional equality) and the initial condition makes sense.

Let us consider vector fields as derivations, and the functional μ ↦ F_{η}(μ) := ∫ ηdμ, for η ∈ C_{c}^{∞}(ℝ^{d}) fixed. By the continuity equation, the derivative of F_{η} along u_{t}^{2} is equal to ⟨∇η, u_{t}^{2}⟩_{μ_{t}^{2}}, therefore from the fact that u_{0}^{1} = v_{0}^{2} we get:

u^{1}(u^{2}(F_{η}))(μ) = \frac{d}{dt}⟨∇η, u_{t}^{2}⟩_{μ_{t}^{2}}|_{t=0} = ⟨∇^{2}η · v_{0}^{2}, u_{0}^{2}⟩_{μ} + ⟨∇η, \frac{d}{dt}u_{t}^{2}|_{t=0}⟩_{μ}
= ⟨∇^{2}η · u_{0}^{1}, u_{0}^{2}⟩_{μ} + ⟨∇η, ∇_{v_{0}^{2}}u_{t}^{2}⟩_{μ}.

Subtracting the analogous term u^{2}(u^{1}(F_{η}))(μ) and using the symmetry of ∇^{2}η and the identities u_{0}^{i} = v_{0}^{1−i}, i = 0, 1, we get

[u^{1}, u^{2}](F_{η})(μ) = ⟨∇η, ∇_{u_{0}^{1}}u_{t}^{2} − ∇_{u_{0}^{2}}u_{t}^{1}⟩_{μ}.

Given that the set {∇η}_{η∈C_{c}^{∞}} is dense in Tan_{μ}(𝒫_{2}(ℝ^{d})), the above equation characterizes [u^{1}, u^{2}] as:

(6.44)  [u^{1}, u^{2}] = ∇_{u_{0}^{1}}u_{t}^{2} − ∇_{u_{0}^{2}}u_{t}^{1},

which proves the torsion-free identity for the covariant derivative.

In the case of the parallel transport along a flow, considered in Remark 6.34, given that the right composition with X(t, s, ·) is an isometry from L_{μ_{t}}^{2} to L_{μ_{s}}^{2}, it holds

⟨u_{s}^{1}, u_{s}^{2}⟩_{μ_{s}} = ⟨u_{s}^{1} ∘ X(t, s, ·), u_{s}^{2} ∘ X(t, s, ·)⟩_{μ_{t}},

subtracting ⟨u_{t}^{1}, u_{t}^{2}⟩_{μ_{t}}, dividing both terms by s − t and letting s → t we get that the Leibnitz rule holds even using the maps X(t, s, ·):

\frac{d}{dt}⟨u_{t}^{1}, u_{t}^{2}⟩_{μ_{t}} = ⟨\frac{d}{ds}u_{s}^{1} ∘ X(t, s, ·)|_{s=t}, u_{t}^{2}⟩_{μ_{t}} + ⟨u_{t}^{1}, \frac{d}{ds}u_{s}^{2} ∘ X(t, s, ·)|_{s=t}⟩_{μ_{t}},

for any couple of vector fields u_{t}^{i} such that t ↦ u_{t}^{i} ∘ X(s, t, ·) is absolutely continuous for i = 1, 2. From this formula it follows that the parallel transport along any flow X compatible with μ_{t} preserves the scalar product.

Of course, different parallel transports define different covariant derivatives ˜∇_{v_{t}}u_{t} via (6.41): they are expressed by

˜∇_{v_{t}}u_{t} := P_{μ_{t}} (\frac{d}{ds}u_{s} ∘ X(t, s, ·)|_{s=t}) .

Denoting by ṽ_{t} the velocity field of X, we get that the covariant derivative of the vector field u_{t} := ∇φ, φ ∈ C_{c}^{∞}(ℝ^{d}), is given by ˜∇_{v_{t}}∇φ = P_{μ_{t}}(∇^{2}φ · ṽ_{t}). It is easy to check that a generic covariant derivative is not torsion-free. Indeed, assume that it is and observe that in this case the two following equations hold:

\begin{aligned}⟨∇_{∇φ^{1}}∇φ^{2}, ∇φ^{3}⟩_{μ} + ⟨∇φ^{2}, ∇_{∇φ^{1}}∇φ^{3}⟩_{μ} & = ⟨ ˜∇_{∇φ^{1}}∇φ^{2}, ∇φ^{3}⟩_{μ} + ⟨∇φ^{2}, ˜∇_{∇φ^{1}}∇φ^{3}⟩_{μ}, \\ ∇_{∇φ^{1}}∇φ^{2} − ∇_{∇φ^{2}}∇φ^{1} & = ˜∇_{∇φ^{1}}∇φ^{2} − ˜∇_{∇φ^{2}}∇φ^{1}, \end{aligned}

for any φ^{i} ∈ C_{c}^{∞}(ℝ^{d}), i = 1, 2, 3. From these equalities, with some algebraic manipulations (more explicitely, by following the calculations indicated in the Koszul formula), it follows that ⟨∇_{∇φ^{1}}∇φ^{2}, ∇φ^{3}⟩_{μ} = ⟨ ˜∇_{∇φ^{1}}∇φ^{2}, ∇φ^{3}⟩_{μ}, so that the two covariant derivatives coincide.

It remains to prove that if a different flow X induces the same covariant derivative of the flow T, then X = T. To prove this observe that in the identity

⟨∇^{2}φ · ṽ_{t}, ∇η⟩_{μ_{t}} = ⟨∇^{2}φ · v_{t}, ∇η⟩_{μ_{t}} ∀φ, η ∈ C_{c}^{∞}(ℝ^{d})

we can use test functions φ ∈ C^{2}(ℝ^{d}) with uniformly bounded second derivatives (by a simple approximation argument based on the finiteness of the second moments of μ_{t}). Choosing φ(x) = |⟨x, ξ⟩|^{2} gives

∫ \frac{∂η}{∂ξ} ⟨ṽ_{t} − v_{t}, ξ⟩dμ_{t} = 0 ∀η ∈ C_{c}^{∞}(ℝ^{d}), ξ ∈ ℝ^{d}.

This means the symmetric part of the distributional derivative of the vector-valued distribution (ṽ_{t} −v_{t})μ_{t} vanishes; Korn’s inequality gives that the distribution is equivalent to a constant. By integrability, this constant must be 0, i.e. ṽ_{t} = v_{t} μ_{t}-a.e. in ℝ^{d}.

The definition of covariant derivative allows us to define the curvature tensor and to check, at least formally, that (𝒫_{2}(ℝ^{d}), W_{2}) is positively curved by proving that its sectional curvatures are always non-negative. The spirit of the foregoing discussion and the calculations we do, are basically borrowed from Lott’s work [43].

Given four vector fields μ ↦ ∇φ_{μ}^{i} ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})), i = 1, …, 4, the curvature tensor R calculated on them is defined as:

⟨R(∇φ_{μ}^{1}, ∇φ_{μ}^{2})(∇φ_{μ}^{3}), ∇φ_{μ}^{4}⟩_{μ} : = ⟨∇_{∇φ_{μ}^{1}}(∇_{∇φ_{μ}^{2}}∇φ_{μ}^{3}), ∇φ_{μ}^{4}⟩_{μ}
− ⟨∇_{∇φ_{μ}^{2}}(∇_{∇φ_{μ}^{1}}∇φ_{μ}^{3}), ∇φ_{μ}^{4}⟩_{μ}
− ⟨∇_{[∇φ_{μ}^{1},∇φ_{μ}^{2}]}∇φ_{μ}^{3}, ∇φ_{μ}^{4}⟩_{μ}.

With the same calculation used in the classical Riemannian case, it is easy to check that R is actually a tensor, i.e. that its value at the measure μ depends only on the value of the four vector fields at μ. Therefore in order to evaluate it, we can consider the simpler vector fields μ ↦ ∇φ^{i} ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})), i = 1, …, 4, where the functions φ^{i} do not depend on the base measure μ. This will simplify the calculations. Under this assumption we have

(6.45)  ∇_{v}∇φ = P_{μ}(∇^{2}φ · v)  ∀v ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})).

In order to give an explicit formula for R, it is useful to introduce the function ξ_{μ}(φ^{1}, φ^{2}) ∈ L_{μ}^{2} as

ξ_{μ}(φ^{1}, φ^{2}) := P_{μ}^{⊥}(∇^{2}φ^{1} · ∇φ^{2}) = ∇^{2}φ^{1} · ∇φ^{2} − ∇_{∇φ^{2}}∇φ^{1}(μ).

Observe that from ∇^{2}φ^{1} · ∇φ^{2} + ∇^{2}φ^{2} · ∇φ^{1} = ∇(⟨∇φ^{1}, ∇φ^{2}⟩) ∈ Tan_{μ}(𝒫_{2}(ℝ^{d})) we get ξ_{μ}(φ^{1}, φ^{2}) = −ξ_{μ}(φ^{2}, φ^{1}).

Proposition 6.44. The curvature tensor is given by

\begin{matrix}⟨R(∇φ^{1}, ∇φ^{2})(∇φ^{3}), ∇φ^{4}⟩_{μ} = & ⟨ξ_{μ}(φ^{1}, φ^{4}), ξ_{μ}(φ^{2}, φ^{3})⟩_{μ} − ⟨ξ_{μ}(φ^{1}, φ^{3}), ξ_{μ}(φ^{2}, φ^{4})⟩_{μ} \\ − 2⟨ξ_{μ}(φ^{1}, φ^{2}), ξ_{μ}(φ^{3}, φ^{4})⟩_{μ}. \end{matrix}

Proof. Define μ_{t} := (Id + t∇φ^{1})_{#}μ and F (ν) := ∫ ηdν with η := ⟨∇^{2}φ^{3} · ∇φ^{2}, ∇φ^{4}⟩. Evaluate the derivative at t = 0 of F (μ_{t}) to get

\frac{d}{dt}F (μ_{t})|_{t=0} = \frac{d}{dt} ∫ η ∘ (Id + t∇φ^{1})dμ|_{t=0} = ⟨∇η, ∇φ^{1}⟩_{μ}.

On the other hand, using equations (6.45) and (6.43) we have

\frac{d}{dt}F (μ_{t})|_{t=0} = \frac{d}{dt}⟨∇^{2}φ^{3} · ∇φ^{2}, ∇φ^{4}⟩_{μ_{t}}|_{t=0}

= \frac{d}{dt}⟨∇_{∇φ^{2}}∇φ^{3}(μ_{t}), ∇φ^{4}⟩_{μ_{t}}|_{t=0}

= ⟨∇_{∇φ^{1}}(∇_{∇φ^{2}}∇φ^{3}), ∇φ^{4}⟩_{μ} + ⟨∇_{∇φ^{2}}∇φ^{3}, ∇_{∇φ^{1}}∇φ^{4}⟩_{μ}.

Coupling the last two equations and then using the trivial identity ⟨P_{μ}(v), P_{μ}(w)⟩_{μ} = ⟨v, w⟩_{μ} − ⟨P_{μ}^{⊥}(v), P_{μ}^{⊥}(w)⟩_{μ}, valid for any v, w ∈ L_{μ}^{2}, we obtain the equality

⟨∇_{∇φ^{1}}(∇_{∇φ^{2}}∇φ^{3}), ∇φ^{4}⟩_{μ} = ⟨∇(⟨∇^{2}φ^{3} · ∇φ^{2}, ∇φ^{4}⟩), ∇φ^{1}⟩_{μ} − ⟨∇_{∇φ^{2}}∇φ^{3}, ∇_{∇φ^{1}}∇φ^{4}⟩_{μ}

= ⟨∇(⟨∇^{2}φ^{3} · ∇φ^{2}, ∇φ^{4}⟩), ∇φ^{1}⟩_{μ} − ⟨∇^{2}φ^{3} · ∇φ^{2}, ∇^{2}φ^{4} · ∇φ^{1}⟩_{μ}

+ ⟨ξ_{μ}(φ^{3}, φ^{2}), ξ_{μ}(φ^{4}, φ^{1})⟩_{μ}.

The computation of the gradient of ⟨∇^{2}φ^{3} · ∇φ^{2}, ∇φ^{4}⟩ gives

\begin{aligned}⟨∇_{∇φ^{1}}(∇_{∇φ^{2}}∇φ^{3}), ∇φ^{4}⟩_{μ} & = ∫ ∇^{3}φ^{3}(∇φ^{2}, ∇φ^{4}, ∇φ^{1})dμ + ⟨∇^{2}φ^{3} · ∇φ^{4}, ∇^{2}φ^{2} · ∇φ^{1}⟩_{μ} \\ & + ⟨ξ_{μ}(φ^{3}, φ^{2}), ξ_{μ}(φ^{4}, φ^{1})⟩_{μ}. \end{aligned} (6.46)

Analogously, it holds:

\begin{aligned}⟨∇_{∇φ^{2}}(∇_{∇φ^{1}}∇φ^{3}), ∇φ^{4}⟩_{μ} & = ∫ ∇^{3}φ^{3}(∇φ^{1}, ∇φ^{4}, ∇φ^{2})dμ + ⟨∇^{2}φ^{3} · ∇φ^{4}, ∇^{2}φ^{1} · ∇φ^{2}⟩_{μ} \\ & + ⟨ξ_{μ}(φ^{3}, φ^{1}), ξ_{μ}(φ^{4}, φ^{2})⟩_{μ}, \end{aligned} (6.47)

so that, subtracting (6.47) from (6.46), the symmetry of ∇^{3}φ^{3} gives

\begin{matrix}⟨∇_{∇φ^{1}}(∇_{∇φ^{2}}∇φ^{3}), ∇φ^{4}⟩_{μ} − ⟨∇_{∇φ^{2}}(∇_{∇φ^{1}}∇φ^{3}), ∇φ^{4}⟩_{μ} \\ = ⟨∇^{2}φ^{3} · ∇φ^{4}, ∇^{2}φ^{2} · ∇φ^{1}⟩_{μ} − ⟨∇^{2}φ^{3} · ∇φ^{4}, ∇^{2}φ^{1} · ∇φ^{2}⟩_{μ} \\ + ⟨ξ_{μ}(φ^{3}, φ^{2}), ξ_{μ}(φ^{4}, φ^{1})⟩_{μ} − ⟨ξ_{μ}(φ^{3}, φ^{1}), ξ_{μ}(φ^{4}, φ^{2})⟩_{μ}. \end{matrix} ; (6.48)

Recalling equation (6.44) we get

⟨∇_{[∇φ^{1},∇φ^{2}]}∇φ^{3}, ∇φ^{4}⟩_{μ} = ⟨∇^{2}φ^{3} · P_{μ}(∇^{2}φ^{2} · ∇φ^{1} − ∇^{2}φ^{1} · ∇φ^{2}), ∇φ^{4}⟩_{μ}
= ⟨P_{μ}(∇^{2}φ^{2} · ∇φ^{1} − ∇^{2}φ^{1} · ∇φ^{2}), ∇^{2}φ^{3} · ∇φ^{4}⟩_{μ}
= ⟨∇^{2}φ^{3} · ∇φ^{4}, ∇^{2}φ^{2} · ∇φ^{1} − ∇^{2}φ^{1} · ∇φ^{2}⟩_{μ}
− ⟨ξ_{μ}(φ^{2}, φ^{1}), ξ_{μ}(φ^{3}, φ^{4})⟩_{μ} + ⟨ξ_{μ}(φ^{1}, φ^{2}), ξ_{μ}(φ^{3}, φ^{4})⟩_{μ}.

Subtracting the last equations from (6.48), all the terms except those involving the functions ξ_{μ} cancel, and the thesis follows. □

From the representation formula of the curvature tensor, it follows immediately that the sectional curvatures of 𝒫_{2}(ℝ^{d}) are non-negative (for the definition see, for instance, [33], Chapter 4, section 3). Indeed, it holds:

K(∇φ, ∇ψ)(μ) = \frac{⟨R(∇φ, ∇ψ)∇ψ, ∇φ⟩_{μ}}{∥∇φ∥_{μ}^{2}∥∇ψ∥_{μ}^{2} − ⟨∇φ, ∇ψ⟩_{μ}^{2}} = \frac{3∥ξ_{μ}(φ^{1}, φ^{2})∥_{μ}^{2}}{∥∇φ∥_{μ}^{2}∥∇ψ∥_{μ}^{2} − ⟨∇φ, ∇ψ⟩_{μ}^{2}} ≥ 0.

6. A distance on the tangent bundle

Recall that, for a Riemannian manifold M, it is possible to endow the tangent bundle T M with a natural Riemannian metric, the so-called Sasaki metric, in the following way (see also [33], Chapter 3, exercise 2).

Fix a point (p, u) ∈ T M and choose two regular curves [0, 1] ∋ t → α^{i}(t) ∈ T M, i = 1, 2, such that α^{1}(0) = α^{2}(0) = (p, u). Let (p^{i}(t), u^{i}(t)) := α^{i}(t) and v^{i}(t) := (p^{i}(t))′, i = 1, 2. Clearly V^{i} := (α^{i})′(0) ∈ T_{(p,u)}(T M), i = 1, 2. The scalar product ⟨·, ·⟩^{∗} between V^{1} and V^{2} is defined as

⟨V^{1}, V^{2}⟩^{∗} := ⟨v^{1}(0), v^{2}(0)⟩ + ⟨∇_{v^{1}}u^{1}(0), ∇_{v^{2}}u^{2}(0)⟩.

It is possible to show that this is a good definition, that is, it depends only on V^{1}, V^{2} and not on the particular curves α^{1}(t), α^{2}(t), therefore it defines a metric tensor on T M. It is then easy to see that the distance d on T M induced by this metric tensor is given by

(6.49)  d^{2}((p^{1}, u^{1}), (p^{2}, u^{2})) = inf_{γ} (L(γ))^{2} + |T (u^{1}) − u^{2}|^{2},

where the infimum is taken among all the smooth curves γ(t) in M connecting p^{1} to p^{2}, L(γ) is the length of γ and T (u^{1}) is the parallel transport of u^{1} along γ to the point p^{2}. This definition has a Wasserstein analogous, as it doesn’t rely on a differentiable structure of the tangent bundle to be written.

So turn back to the space (𝒫_{2}(ℝ^{d}), W_{2}) and recall that for a given absolutely continuous curve t → μ_{t} on [0, 1] we define L_{t_{1}}^{t_{2}}(μ_{t}), 0 ≤ t_{1} ≤ t_{2} ≤ 1, as the length of the curve on the interval [t_{1}, t_{2}]: L_{t_{1}}^{t_{2}}(μ_{t}) := ∫_{t_{1}}^{t_{2}} ∥v_{t}∥_{μ_{t}}dr, where v_{t} is the velocity vector field of μ_{t}. We will also set L(μ_{t}) := L_{0}^{1}(μ_{t}) for a curve parametrized in [0, 1]. Recall also that for a regular curve t → μ_{t}, we define T_{t}^{s} : Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})) → Tan_{μ_{s}}(𝒫_{2}(ℝ^{d})) as the parallel transport maps.

The tangent bundle is defined as

Tan (𝒫_{2}(ℝ^{d})) := {(μ, u) : μ ∈ 𝒫_{2}(ℝ^{d}), u ∈ Tan_{μ}(𝒫_{2}(ℝ^{d}))} .

Throughout all this section, we say that a sequence (μ_{n}, u_{n}) converges to (μ, u) if the measures converge w.r.t. W and the maps converge strongly, i.e. if:

lim_{n→∞} W (μ_{n}, μ) = 0,

lim_{n→∞} ∥u_{n}∥_{μ_{n}} = ∥u∥_{μ},

lim_{n→∞} ⟨u_{n}, ∇φ⟩_{μ_{n}} = ⟨u, ∇φ⟩_{μ} ∀φ ∈ C_{c}^{∞}(ℝ^{d}, ℝ).

(6.50)

Remark 6.45. Actually in general, to have strong convergence of maps we should ask for the third of the above equations to hold for any smooth vector valued map η ∈ C^{∞}(ℝ^{d}, ℝ^{d}), and not just for gradients. However, since the vectors (u_{n}), u are tangent, this more restrictive choice is still sufficient to get the desired convergence. Indeed, let \overline{u} be any weak limit of (u_{n}) (which exists, because the second equation gives a uniform bound on the norms), and observe that the third equation implies P_{μ}(\overline{u}) = u, therefore the claim follows from

∥u∥_{μ} ≤ ∥\overline{u}∥_{μ} ≤ \underline{lim}_{n→∞} ∥u_{n}∥_{μ_{n}} = ∥u∥_{μ}.

In the following we will, with some abuse of notation, say u_{n} converges to u instead than (μ_{n}, u_{n}) converge to (μ, u).

By analogy with equation (6.49), we define the following function on [Tan (𝒫_{2}(ℝ^{d}))]^{2}:

d^{2}((μ, u), (ν, v)) := inf {(L_{0}^{1}(μ_{t}))^{2} + ∥v − T_{0}^{1}(u)∥_{ν}^{2}},

where the infimum is taken on the set of regular curves t → μ_{t} on [0, 1] such that μ_{0} = μ and μ_{1} = ν. In particular we define d((μ, u), (ν, v)) := +∞ if there is no regular curve connecting μ and ν.

The first properties of d are collected below.

Proposition 6.46. The function d is non-negative, symmetric and satisfies the triangular inequality. Moreover for any couple (μ, u), (ν, v) ∈ Tan (𝒫_{2}(ℝ^{d})) and every φ ∈ C_{c}^{∞}(ℝ^{d}) it holds:

\begin{aligned}W (μ, ν) & ≤ d((μ, u), (ν, v)), \\ ∥|u∥_{μ} − ∥v∥_{ν}| & ≤ d((μ, u), (ν, v)), \\ |⟨u, ∇φ⟩_{μ} − ⟨v, ∇φ⟩_{ν}| & ≤ d((μ, u), (ν, v))(∥u∥_{μ} Lip(∇φ) + ∥∇φ∥_{ν}). \end{aligned}

Proof. The non-negativity and the triangle property are clear. For the symmetry it is enough to observe that for any regular curve connecting μ and ν it holds ∥T_{0}^{1}(u)−v∥_{ν} = ∥u − T_{1}^{0}(v)∥_{μ}, given that the parallel transport is norm preserving.

The first inequality is obvious. For the second, notice that ∥u∥_{μ} = ∥T_{0}^{1}(u)∥_{ν}, therefore ∥|u∥_{μ} − ∥v∥_{ν}| = |∥T_{0}^{1}(u)∥_{ν} − ∥v∥_{ν}| ≤ ∥T_{0}^{1}(u) − v∥_{ν}. To prove the last one start observing that for any regular curve t → μ_{t} connecting μ to ν, the function t → ⟨T_{0}^{t}(u), ∇φ⟩_{μ_{t}} is absolutely continuous and its derivative is given by

\frac{d}{dt}⟨T_{0}^{t}(u), ∇φ⟩_{μ_{t}} = ⟨T_{0}^{t}(u), ∇^{2}φ · v_{t}⟩_{μ_{t}} ≤ Lip(∇φ)∥u∥_{μ}∥v_{t}∥_{μ_{t}},

where v_{t} are the velocity vectors of t → μ_{t}. Integrate to obtain

|⟨u, ∇φ⟩_{μ} − ⟨T_{0}^{1}(u), ∇φ⟩_{ν}| ≤ Lip(∇φ)∥u∥_{μ}L(μ_{t}).

Finally observe that

|⟨T_{0}^{1}(u), ∇φ⟩_{ν} − ⟨v, ∇φ⟩_{ν}| ≤ ∥T_{0}^{1}(u) − v∥_{ν}∥∇φ∥_{ν}.
□

The first properties stated above suggest that d behaves like a distance on Tan (𝒫_{2}(ℝ^{d})), the only problem being that it is not real valued. Given that the regular curves are dense in the set of absolutely continuous curves, a natural candidate for a relaxation of d is its lower semicontinuous envelope d^{∗}, defined by:

d^{∗}((μ, u), (ν, v)) := inf_{{(μn,un)→(μ,u) \atop (νn,vn)→(ν,v)}} \underline{lim}_{n→∞} d((μ_{n}, u_{n}), (ν_{n}, v_{n})).

However, it seems that the function d^{∗} does not have sufficient regularity properties. For instance it is not clear - to the author - whether d^{∗} satisfies the triangle inequality. Therefore we modify a bit the definition, and we introduce the function D as:

D((μ, u), (ν, v)) := inf_{φ,ψ∈C_{c}^{∞}(ℝ^{d})}{d^{∗}((μ, ∇φ), (ν, ∇ψ)) + ∥u − ∇φ∥_{μ} + ∥v − ∇ψ∥_{ν}}.

With the introduction of D we are allowed to regularize the vectors u, v provided we pay the L^{2} difference between the regularizations and the vectors themselves.

We want to prove that D is a distance on Tan (𝒫_{2}(ℝ^{d})) with metrizes the convergence of maps.

It is easy to check that the following inequalities are valid for any couple (μ, u), (ν, v) ∈ Tan (𝒫_{2}(ℝ^{d})) and every φ ∈ C_{c}^{∞}:

\begin{aligned}W (μ, ν) & ≤ d^{∗}((μ, u), (ν, v)), \\ |∥u∥_{μ} − ∥v∥_{ν}| & ≤ d^{∗}((μ, u), (ν, v)), \\ |⟨u, ∇φ⟩_{μ} − ⟨v, ∇φ⟩_{ν}| & ≤ d^{∗}((μ, u), (ν, v))(∥u∥_{μ} Lip(∇φ) + ∥∇φ∥_{ν}). \end{aligned}

Indeed all of terms, apart d^{∗}, are continuous w.r.t. to convergence in Tan (𝒫_{2}(ℝ^{d})): therefore the inequalities follow directly from the analogous inequalities valid for d.

From the latter inequalities it is just a matter of simple algebraic manipulations to show that D satisfies:

(6.50a)  W (μ, ν) ≤ D((μ, u), (ν, v)),
(6.50b)  |∥u∥_{μ} − ∥v∥_{ν}| ≤ D((μ, u), (ν, v)),

|⟨u, ∇φ⟩_{μ} − ⟨v, ∇φ⟩_{ν}| ≤ D((μ, u), (ν, v))(∥u∥_{μ} Lip(∇φ) + 2∥∇φ∥_{ν} + ∥∇φ∥_{μ}). (6.50c)

Proposition 6.47 (D is a distance). The function D is a distance on Tan (𝒫_{2}(ℝ^{d})).

Proof. Clearly D is non-negative, symmetric and D((μ, u), (μ, u)) = 0. Equations (6.50) show that if D((μ, u), (ν, v)) = 0, then μ = ν and u = v.

Consider now a geodesic μ_{t} parametrized in [0, 1] connecting μ to ν and find a vector field (not necessarly continuous) u_{t} such that (μ_{t}, u_{t}) converges to (μ, u) as t → 0 and to (ν, v) as t → 1. Recalling that any geodesic is regular if restricted to the interval [ε, 1 − ε] we can bound d^{∗} from above with:

d^{∗}((μ, u), (ν, v)) ≤ \underline{lim}_{ε→0} d((μ_{ε}, u_{ε}), (μ_{1−ε}, u_{1−ε}))

≤ \underline{lim}_{ε→0} W (μ, ν) + ∥T_{ε}^{1−ε}(u_{ε}) − u_{1−ε}∥_{μ_{1−ε}}

≤ W (μ, ν) + ∥u∥_{μ} + ∥v∥_{ν}.

Being d^{∗} real valued, so is D.

It remains to prove the triangle inequality. Fix three couples (μ^{i}, u^{i}), i = 1, 2, 3, and choose a minimizing sequences for D((μ^{1}, u^{1}), (μ^{2}, u^{2})), that is: two sequences of smooth functions u_{n}^{1}, u_{n}^{2,a} ∈ C_{c}^{∞}(ℝ^{d}, ℝ), n ∈ ℕ, and, for every n ∈ ℕ, two sequences of measures (μ_{n,m}^{1}), (μ_{n,m}^{2,a}), and a sequence of regular curves t → ν_{n,m}^{a}(t) connecting μ_{n,m}^{1} to μ_{n,m}^{2,a}, such that: lim_{m→∞} W (μ_{n,m}^{1}, μ^{1}) = 0, lim_{m→∞} W (μ_{n,m}^{2,a}, μ^{2}) = 0 and

d^{∗}((μ^{1}, ∇u_{n}^{1}), (μ^{2}, ∇u_{n}^{2,a})) = lim_{m→∞} (L^{2}(ν_{n,m}^{a}) + ∥(T_{n,m}^{a})_{0}^{1}(∇u_{n}^{1}) − ∇u_{n}^{2,a}∥_{μ_{n,m}^{2,a}}^{2})^{1/2},
D((μ^{1}, u^{1}), (μ^{2}, u^{2})) = lim_{n→∞} (d^{∗}((μ^{1}, ∇u_{n}^{1}), (μ^{2}, ∇u_{n}^{2}))
+ ∥∇u_{n}^{1} − u^{1}∥_{μ^{1}} + ∥∇u_{n}^{2,a} − u^{2}∥_{μ^{2}}),

where (T_{n,m}^{a})_{t}^{s}, t, s ∈ [0, 1], are the parallel transport maps along t → ν_{n,m}^{a}(t). It is easy to check, using proposition 6.21, that the curves t → ν_{n,m}^{a}(t) may be chosen such that for every n, m the support of the measure ν_{n,m}^{a}(t) is made of exactly m points, each of mass 1/m. Pick an analogous minimizing sequence for D((μ^{2}, u^{2}), (μ^{3}, u^{3})), made of smooth functions u_{n}^{2,b}, u_{n}^{3}, measures μ_{n,m}^{2,b}, μ_{n,m}^{3} and curves t → ν_{n,m}^{b}(t) from μ_{n,m}^{2,b} to μ_{n,m}^{3}. Observe that W (μ_{n,m}^{2,a}, μ_{n,m}^{2,b}) → 0 as m → ∞, because both the sequences converge to μ^{2}, and that for every n, m, both the measures μ_{n,m}^{2,a} and μ_{n,m}^{2,b} are discrete and the mass of their points is 1/m. In this case we know (see also remark 1.16) that there exists an optimal transport map from μ_{n,m}^{2,a} to μ_{n,m}^{2,b}. Let [0, 1] ∋ t → σ_{n,m}(t) be the geodesic induced by this map and observe that it is regular on the whole [0, 1].

Define a regular curve t → μ_{n,m}(t) on [0, 3] by joining ν_{n,m}^{a}(t), σ_{n,m}(t), ν_{n,m}^{b}(t): its extreme points are μ_{n,m}^{1}, μ_{n,m}^{3}, which converge to μ^{1} and μ^{3} as m → ∞. Therefore, for a fixed n we can use these curves to estimate d^{∗}((μ^{1}, ∇u_{n}^{1}), (μ^{3}, ∇u_{n}^{3})) as follows ((T_{n,m})_{t}^{s} are the parallel transport maps along t → μ_{n,m}(t)):

d^{∗}((μ^{1}, ∇u_{n}^{1}), (μ^{3}, ∇u_{n}^{3})) ≤ \underline{lim}_{m→∞} (L^{2}(μ_{n,m}) + ∥(T_{n,m})_{0}^{3}(∇u_{n}^{1}) − ∇u_{n}^{3}∥_{μ_{n,m}^{3}})^{1/2}

≤ \overline{lim}_{m→∞} (L^{2}(ν_{n,m}^{1}) + ∥(T_{n,m})_{0}^{1}(∇u_{n}^{1}) − ∇u_{n}^{2,a}∥_{μ_{n,m}^{2,a}})^{1/2}

+ \overline{lim}_{m→∞} (L^{2}(σ_{n,m}) + ∥(T_{n,m})_{1}^{2}(∇u_{n}^{2,a}) − ∇u_{n}^{2,b}∥_{μ_{n,m}^{2,b}})^{1/2}

+ \overline{lim}_{m→∞} (L^{2}(ν_{n,m}^{2}) + ∥(T_{n,m})_{2}^{3}(∇u_{n}^{2,b}) − ∇u_{n}^{3}∥_{μ_{n,m}^{3}})^{1/2}

≤ d^{∗}((μ^{1}, ∇u_{n}^{1}), (μ^{2}, ∇u_{n}^{2,a})) + d^{∗}((μ^{2}, ∇u_{n}^{2,b}), (μ^{3}, ∇u_{n}^{3}))

+ ∥∇u_{n}^{2,a} − ∇u_{n}^{2,b}∥_{μ^{2}}

≤ d^{∗}((μ^{1}, ∇u_{n}^{1}), (μ^{2}, ∇u_{n}^{2,a})) + d^{∗}((μ^{2}, ∇u_{n}^{2,b}), (μ^{3}, ∇u_{n}^{3}))

+ ∥∇u_{n}^{2,a} − u^{2}∥_{μ^{2}} + ∥∇u^{2} − u_{n}^{2,b}∥_{μ^{2}},

where in the third step we used lemma 6.35 and the fact that W (μ_{n,m}^{2,a}, μ_{n,m}^{2,b}) → 0 as m → ∞.

Now choose ∇u_{n}^{1}, ∇u_{n}^{3} as regular functions to bound from above D((μ^{1}, u^{1}), (μ^{3}, u^{3})) with

D((μ^{1}, u^{1}), (μ^{3}, u^{3}))

≤ \underline{lim}_{n→∞} (d^{∗}((μ^{1}, ∇u_{n}^{1}), (μ^{3}, ∇u_{n}^{3})) + ∥u^{1} − ∇u_{n}^{1}∥_{μ^{1}} + ∥u^{3} − ∇u_{n}^{3}∥_{μ^{3}})

≤ \underline{lim}_{n→∞} (d^{∗}((μ^{1}, ∇u_{n}^{1}), (μ^{2}, ∇u_{n}^{2,a})) + d^{∗}((μ^{2}, ∇u_{n}^{2,b}), (μ^{3}, ∇u_{n}^{3}))

+ ∥∇u_{n}^{2,a} − u^{2}∥_{μ^{2}} + ∥∇u^{2} − u_{n}^{2,b}∥_{μ^{2}} + ∥u^{1} − ∇u_{n}^{1}∥_{μ^{1}} + ∥u^{3} − ∇u_{n}^{3}∥_{μ^{3}})

≤ D((μ^{1}, u^{1}), (μ^{2}, u^{2})) + D((μ^{2}, u^{2}), (μ^{3}, u^{3})).

□

Proposition 6.48 (D metrizes the convergence in Tan (𝒫_{2}(ℝ^{d}))). A sequence (μ_{n}, u_{n}) converges to (μ, u) in Tan (𝒫_{2}(ℝ^{d})) in the sense of equations (6.50) if and only if D((μ_{n}, u_{n}), (μ, u)) converges to 0.

Proof. Let (μ, u), (μ_{n}, u_{n}), n ∈ ℕ, be given couples in Tan (𝒫_{2}(ℝ^{d})).

Suppose that D((μ, u), (μ_{n}, u_{n})) → 0. Then equation (6.50a) implies that W (μ, μ_{n}) → 0, equation (6.50c) that the maps u_{n} converge weakly to u and equation (6.50b) gives the convergence of norms. Therefore the maps converge in Tan (𝒫_{2}(ℝ^{d})).

For the converse implication, fix a function φ ∈ C_{c}^{∞}(ℝ^{d}) and choose, for every n, a geodesic μ_{n}(t) from μ to μ_{n} and let, for any 0 < ε < 1, T_{n}^{ε} be the parallel transport map from μ_{n}(ε) to μ_{n}(1 − ε). Fix n ∈ ℕ and observe that we have the following bound:

D((μ, u), (μ_{n}, u_{n})) ≤ d^{∗}((μ, ∇φ), (μ_{n}, ∇φ)) + ∥u − ∇φ∥_{μ} + ∥u_{n} − ∇φ∥_{μ_{n}}

≤ lim_{ε→0} ((1 − 2ε)^{2}W^{2}(μ, μ_{n}) + ∥T_{n}^{ε}(∇φ) − ∇φ∥_{μ_{n}(1−ε)})^{1/2}

+ ∥u − ∇φ∥_{μ} + ∥u_{n} − ∇φ∥_{μ_{n}}

≤ W (μ, μ_{n}) + W (μ, μ_{n}) Lip(∇φ) + ∥u − ∇φ∥_{μ} + ∥u_{n} − ∇φ∥_{μ_{n}},

having used lemma 6.35. Letting n → ∞ we get

\overline{lim}_{n→∞} D((μ, u), (μ_{n}, u_{n})) ≤ 2∥u − ∇φ∥_{μ},

therefore the thesis follows by letting ∇φ → u in L_{μ}^{2}. □

Remark 6.49. With the same spirit it is possible to extend D to a function on Tan (𝒫_{2}(ℝ^{d})) := {(μ, γ) : γ ∈ 𝒫_{2}(ℝ^{2d})_{μ}}, provided we define what do we mean for convergence of a sequence of maps u_{n} ∈ Tan_{μ_{n}}(𝒫_{2}(ℝ^{d})) to a plan γ ∈ 𝒫_{2}(ℝ^{2d})_{μ}. However, independently from this definition, it is possibile to show that the resulting space (Tan (𝒫_{2}(ℝ^{d})), D) is not the completion of the space (Tan (𝒫_{2}(ℝ^{d})), D). Indeed, the problem is that a sequence of tangent maps may converge strongly to a non tangent map: in proposition 6.11 we showed that actually any vector, tangent or not, may be approximated in the sense of strong convergence of maps with tangent vectors.

The distance D is closely related to the behavior of parallel transport along regular curves, as the following proposition shows.

Proposition 6.50. Let t → (μ_{t}, u_{t}) be an absolutely continuous curve in (Tan (𝒫_{2}(ℝ^{d})), D). Then t → μ_{t} is an absolutely continuos curve in (𝒫_{2}(ℝ^{d}), W) and it holds

(6.51)  lim_{s→t} \frac{D((μ_{s}, u_{s}), (μ_{t}, u_{t}))}{|s − t|} ≥ ∥v_{t}∥_{μ_{t}},

for a.e. t, where the v_{t} are the velocity vectors of μ_{t}.

Conversely, if t → μ_{t} is a regular curve and t → u_{t} a parallel transport along it, then t → (μ_{t}, u_{t}) is absolutely continuous in (Tan (𝒫_{2}(ℝ^{d})), D) and the equality holds in the above limit for a.e. t.

Proof. The first claim follows directly from inequality (6.50a). For the second one just observe that D ≤ d and estimate d((μ_{s}, u_{s}), (μ_{t}, u_{t})) from above by using the regular curve r → μ_{(1−r)t+rs}. □

This proposition suggests a way to generalize the definition of parallel transport in the following way:

Definition 6.51 (Weak parallel transport). Let t → μ_{t} be an absolutely continuous curve and v_{t} its velocity vectors. We say that t → u_{t} ∈ Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})) is a weak parallel transport along μ_{t} if t → (μ_{t}, u_{t}) is absolutely continuous in (Tan (𝒫_{2}(ℝ^{d})), D) and equation (6.51) holds for a.e. t.

For the weak parallel transport we are able to prove that it is norm preserving and satisfies equation (6.22), which characterizes the parallel transport along regular curves without the flow maps.

Proposition 6.52 (Properties of weak parallel transport). Let μ_{t} be an absolutely continuous curve and let u_{t} ∈ Tan_{μ_{t}}(𝒫_{2}(ℝ^{d})) be a weak parallel transport along it. Then t → ∥u_{t}∥_{μ_{t}} is constant, t → ⟨u_{t}, ∇η⟩_{μ_{t}} is absolutely continuous for every η ∈ C_{c}^{∞}(ℝ^{d}) and it holds

\frac{d}{dt}⟨u_{t}, ∇η⟩_{μ_{t}} = ⟨u_{t}, ∇^{2}η · v_{t}⟩_{μ_{t}}, a.e. t.

Proof. Start observing that from inequality (6.50b) and the absolute continuity of t → (μ_{t}, u_{t}) ∈ (Tan (𝒫_{2}(ℝ^{d})), D) it follows that t → ∥u_{t}∥_{μ_{t}} is absolutely continuous, too.

Now let Σ ⊂ [0, 1] be the set of t such that the metric derivative |μ̇_{t}| and the velocity vector v_{t} of μ_{t} are defined and the equation of weak parallel transport is satisfied, i.e. the set of those t ∈ [0, 1] such that:

lim_{h→0} \frac{∫ φdμ_{t+h} − ∫ φdμ_{t}}{h} = ⟨v_{t}, ∇φ⟩_{μ_{t}}, ∀φ ∈ C_{c}^{∞}(ℝ^{d}),
lim_{h→0} \frac{W (μ_{t+h}, μ_{t})}{h} = ∥v_{t}∥_{μ_{t}},
lim_{h→0} \frac{D((μ_{t}, u_{t}), (μ_{t+h}, u_{t+h}))}{h} = ∥v_{t}∥_{μ_{t}},

and recall that by theorem 3.4 we know that L^{1}([0, 1] \ Σ) = 0. Therefore to prove that t → ∥u_{t}∥_{μ_{t}}^{2} is constant, it is sufficient to show that its derivative is 0 at any t ∈ Σ.

Fix such a t and a sequence h^{n} ↓ 0. By definition of D and with a diagunalization argument, we can find first two sequences of functions (φ^{n}), (ψ^{n}) ⊂ C_{c}^{∞}(ℝ^{d}) then a sequence of regular curves parametrized by arc length [0, k^{n}] ∋ r → ν_{r}^{n} such that it holds:

(6.52)  \overline{lim}_{n→∞} \frac{W (μ_{t}, ν_{0}^{n})(1 + Lip(∇φ^{n}))}{h^{n}} = 0,
(6.53)  \overline{lim}_{n→∞} \frac{W (μ_{t+h^{n}}, ν_{k^{n}}^{n})(1 + Lip(∇ψ^{n}))}{h^{n}} = 0,

and

\begin{matrix}D((μ_{t}, u_{t}), (μ_{t+h^{n}}, u_{t+h^{n}})) ≤ & \sqrt{(k^{n})^{2}+∥(T^{n})_{0}^{k^{n}}(∇φ^{n})−∇ψ^{n}∥_{ν_{1}^{n}}^{2}} \\ + ∥∇φ^{n} − u_{t}∥_{μ_{t}} +∥∇ψ^{n} − u_{t+h^{n}}∥_{μ_{t+h^{n}}} \\ ≤ & D((μ_{t}, u_{t}), (μ_{t+h^{n}}, u_{t+h^{n}})) + h^{n}w(n), \end{matrix} ; (6.54)

130 6. AN APPROACH TO SECOND ORDER ANALYSIS

where (T^{n})_{r_{1}}^{r_{2}} are the parallel transport maps along ν_{r}^{n} and w(n) → 0 as n → ∞.

From equations (6.52), (6.53), inequality k^{n} ≥ W (ν_{0}^{n}, ν_{k^{n}}^{n}) and by our choice of t, it follows \underline{lim}_{n} k^{n}/h^{n} ≥ ∥v_{t}∥_{μ_{t}}. Therefore inequalities (6.54) imply

sup_{n} {∥∇ψ∥_{μ_{t+h^{n}}} , ∥∇ψ∥_{ν_{k^{n}}^{n}} , ∥∇φ∥_{μ_{t}}, ∥∇φ∥_{ν_{0}^{n}}} < ∞,
lim_{n→∞} \frac{k^{n}}{h^{n}} = ∥v_{t}∥_{μ_{t}},
lim_{n→∞} \frac{∥(T^{n})_{0}^{k^{n}}(∇φ^{n}) − ∇ψ^{n}∥_{ν_{1}^{n}}}{h^{n}} = 0,
lim_{n→∞} \frac{∥∇φ^{n} − u_{t}∥_{μ_{t}}}{h^{n}} = 0,
lim_{n→∞} \frac{∥∇ψ^{n} − u_{t+h^{n}}∥_{μ_{t+h^{n}}}}{h^{n}} = 0.

(6.55)

Now define A_{n} := ∥∇φ^{n}−u_{t}∥_{μ_{t}}, B_{n} := ∥∇ψ^{n}−u_{t+h^{n}}∥_{μ_{t+h^{n}}}, C_{n} := ∥(T^{n})_{0}^{k^{n}}(∇φ^{n})− ∇ψ^{n}∥_{ν_{1}^{n}}^{2} and observe that it holds

∥|u_{t+h^{n}}∥_{μ_{t+h^{n}}}^{2} − ∥u_{t}∥_{μ_{t}}^{2}| =|⟨u_{t+h^{n}} + ∇ψ^{n}, u_{t+h^{n}} − ∇ψ^{n}⟩_{μ_{t+h^{n}}}
+ ∥∇ψ^{n}∥_{μ_{t+h^{n}}}^{2} − ∥∇ψ^{n}∥_{ν_{k^{n}}^{n}}^{2}
+ ⟨∇ψ^{n} − (T^{n})_{0}^{k^{n}}(∇φ^{n}), ∇ψ^{n} + (T^{n})_{0}^{k^{n}}(∇φ^{n})⟩_{ν_{k^{n}}^{n}}
+ ∥∇φ^{n}∥_{ν_{0}^{n}}^{2} − ∥∇φ^{n}∥_{μ_{t}}^{2}
− ⟨u_{t} + ∇φ^{n}, u_{t} − ∇φ^{n}⟩_{μ_{t}} |
≤B_{n}∥|u_{t+h^{n}}∥_{μ_{t+h^{n}}} + ∥∇ψ^{n}∥_{μ_{t+h^{n}}} |

+ Lip(∇ψ^{n})W (μ_{t+h^{n}}, ν_{k^{n}}^{n})(∥∇ψ^{n}∥_{μ_{t+h^{n}}} + ∥∇ψ^{n}∥_{ν_{k^{n}}^{n}} )

+ C_{n}∥|∇ψ∥_{ν_{k^{n}}^{n}} + ∥∇φ∥_{ν_{0}^{n}} |
+ Lip(∇φ^{n})W (μ_{t}, ν_{0}^{n})(∥∇φ^{n}∥_{μ_{t}} + ∥∇φ^{n}∥_{ν_{0}^{n}})
+ A_{n}∥|u_{t}∥_{μ_{t}} + ∥∇φ∥_{μ_{t}} |.

Dividing by h^{n}, letting n → ∞ and using equations (6.55), we get that

lim_{n→∞} \frac{∥u_{t+h^{n}}∥_{μ_{t+h^{n}}}^{2} − ∥u_{t}∥_{μ_{t}}^{2}}{h^{n}} = 0.

By the arbitrariness of t ∈ Σ and h_{n} ↓ 0 we conclude.

Now we turn to the study of t → ⟨u_{t}, ∇η⟩_{μ_{t}} for a given η ∈ C_{c}^{∞}(ℝ^{d}). Start observing that from the fact that t → ∥u_{t}∥_{μ_{t}} is constant and by inequality (6.50c) we get that t → ⟨u_{t}, ∇η⟩_{μ_{t}} is absolutely continuous. With the same notation as above and with calculations similar to the ones just done, it is possible to check that it holds

lim_{n→∞} \frac{⟨u_{t+h^{n}}, ∇η⟩_{μ_{t+h^{n}}} − ⟨u_{t}, ∇η⟩_{μ_{t}}}{h^{n}} = lim_{n→∞} \frac{⟨(T^{n})_{0}^{k^{n}}(∇φ^{n}), ∇η⟩_{ν_{k^{n}}^{n}} − ⟨∇φ^{n}, ∇η⟩_{ν_{0}^{n}}}{h^{n}} .

Now observe that the right hand side can be evaluated using equation (6.23):

\begin{matrix}lim_{n→∞} & \frac{⟨(T^{n})_{0}^{k^{n}}(∇φ^{n}), ∇η⟩_{ν_{k^{n}}^{n}} − ⟨∇φ^{n}, ∇η⟩_{ν_{0}^{n}}}{h^{n}} \\ = lim_{n→∞} & \frac{1}{h^{n}} ∫_{0}^{k^{n}} ⟨(T^{n})_{0}^{r}(∇φ^{n}), ∇^{2}η · v_{r}^{n}⟩_{ν_{r}^{n}}dr, \end{matrix}

where v_{r}^{n} ∈ Tan_{ν_{r}^{n}}(𝒫_{2}(ℝ^{d})) are the velocity vectors of r → ν_{r}^{n}. By equation (6.55) it follows that for any r ∈ [0, k^{n}] the sequence ((T^{n})_{0}^{r}(∇φ^{n})) strongly converges to u_{t} as n → ∞. Therefore our thesis will be proved if we show that V^{n} := (h^{n})^{−1} ∫_{0}^{k^{n}} v_{r}^{n} ∘ (T^{n})_{0}^{r}dr ∈ L_{ν_{0}^{n}}^{2} weakly converges to v_{t} as n → ∞. To prove this, fix ξ ∈ C_{c}^{∞}(ℝ^{d}) and observe that from equations (6.55) and the fact that Lip(ξ), Lip(∇ξ) < ∞ it follows that

⟨∇ξ, v_{t}⟩_{μ_{t}} = lim_{n→∞} \frac{∫ ξdμ_{t+h^{n}} − ∫ ξdμ_{t}}{h^{n}} = lim_{n→∞} \frac{∫ ξdν_{k^{n}}^{n} − ∫ ξdν_{0}^{n}}{h^{n}} = lim_{n→∞} ⟨∇ξ, V^{n}⟩_{ν_{0}^{n}}.

Let V be any weak limit of (V^{n}) and observe that the above equations are equivalent to P_{μ_{t}}(V) = v_{t}. To prove that V is tangent, and therefore equal to v_{t}, observe that:

∥v_{t}∥_{μ_{t}} = ∥P_{μ_{t}}(V)∥_{μ_{t}} ≤ ∥V ∥_{μ_{t}} ≤ \underline{lim}_{n→∞} ∥V^{n}∥_{ν_{0}^{n}}

≤ \underline{lim}_{n→∞} \frac{∫_{0}^{k^{n}} ∥v_{r}^{n}∥_{ν_{r}^{n}}dr}{h^{n}} = \underline{lim}_{n→∞} \frac{k^{n}}{h^{n}} = ∥v_{t}∥_{μ_{t}},

hence ∥V ∥_{μ_{t}} = ∥P_{μ_{t}}(V)∥_{μ_{t}} and the thesis is proved. □

It should be noticed that it is not clear - to the author - whether the weak parallel transport preserves the scalar product, nor if it is unique. From a tecnhical point of view, the big obstacle is that, if u_{t}^{1}, u_{t}^{2} are two absolutely continuous vector fields w.r.t. D along the same absolutely continuous curve μ_{t}, it is not clear whether the vector field u_{t}^{1} + u_{t}^{2} is absolutely continuous or not.

Furthermore, the density of regular curves is not enough to gain existence of weak parallel transport through an approximation argument. The key problem is that the space (Tan (𝒫_{2}(ℝ^{d})), D) is not complete. Example 6.37 shows that it might be impossible to extend a (weak) parallel transport “backward” to the initial point of a geodesic.

We conclude with a result regarding the lower semicontinuity of the function d. Given the definition of D and d^{∗}, a natural question is the following: is D((μ, u), (ν, v)) = d^{∗}((μ, u), (ν, v)) = d((μ, u), (ν, v)) on those couples such that d((μ, u), (ν, v)) < +∞? The answer is no, as we are going to show now.

The key fact is the possibility to “separate” the supports on the approximating curves. Observe that it always holds d^{∗} ≤ d, and, if the vectors are smooth, D ≤ d^{∗}, therefore we prove our claim if we build an example in which

d^{∗}((μ, u), (ν, v)) < d((μ, u), (ν, v)),

such that u, v ∈ C_{c}^{∞}(ℝ^{d}, ℝ) and the right hand side is finite.

Examples exist in every dimension, however, for the sake of simplicity, we show an explicit construction only in ℝ^{2}.

Fix two positive numbers a, b. Define A := {−1}×[0, a] ⊂ ℝ^{2}, B := {1}×[0, a] ⊂ ℝ^{2} and μ^{0} = μ^{1} := (2a)^{−1}H^{1}|_{A∪B}, where H^{1} is the 1-dimensional Hausdorff measure. Now define

u^{0}(x) := \{ \begin{matrix}(0, b), & x ∈ A, \\ (0, −b), & x ∈ B, \end{matrix}

and u^{1}(x) := −u^{0}(x).

Clearly u^{i} ∈ Tan_{μ^{i}}(𝒫_{2}(ℝ^{d})) ∩ C_{c}^{∞}(ℝ^{d}, ℝ), i = 0, 1. It is clear that there exists at least one regular curve t → μ_{t} connecting μ^{0} to μ^{1} (consider for instance the constant curve t → μ^{0}), therefore d((μ^{0}, u^{0}), (μ^{1}, u^{1})) < ∞. We claim that for a, b large enough it holds

d^{∗}((μ^{0}, u^{0}), (μ^{1}, u^{1})) < d((μ^{0}, u^{0}), (μ^{1}, u^{1})).

We start with a lower bound for d((μ^{0}, u^{0}), (μ^{1}, u^{1})). Choose a regular curve t → μ_{t}

connecting μ^{0} to μ^{1}, let T(t, s, x) be its flow maps and set T(x) := T(0, 1, x). Since supp(μ^{0}) = supp(μ^{1}) = A ∪ B, it must hold T(A ∪ B) = A ∪ B. Moreover, since T is an homeomorphism of ℝ^{2} into itself and the sets A, B are separated it must be true one of the following: case 1 T(A) = A and T(B) = B,

case 2 T(A) = B and T(B) = A. In case 1, examples 6.40 and 6.42 ensure T_{0}^{1}(u^{0}) = u^{0}, therefore we obtain the bound

((L_{0}^{1}(μ_{t}))^{2} + ∥T_{0}^{1}(u^{0}) − u^{1}∥_{μ^{1}}^{2})^{1/2} ≥ ∥u^{0} − u^{1}∥_{μ^{1}} = 2b.

In case 2, proposition 6.53 below, ensures that the length of t → μ_{t} is bounded from below by a/8: we postpone the proof of this to avoid losing continuity in our discussion - here we just observe that this bound exists because if A and B exchange in a continuous way, then “one has to pass above the other”. Therefore it holds:

((L_{0}^{1}(μ_{t}))^{2} + ∥T_{0}^{1}(u^{0}) − u^{1}∥_{μ^{1}}^{2})^{1/2} ≥ L_{0}^{1}(μ_{t}) ≥ a/8.

In summary, we have the following lower bound on d:

(6.56)  d((μ^{0}, u^{0})(μ^{1}, u^{1})) ≥ min \{\frac{a}{8}, 2b\} .

Now we want to build a sequence of measures (μ_{n}), such that W (μ^{0}, μ_{n}) → 0, and two sequences of functions (u_{n}^{i}) ∈ Tan_{μ_{n}}(𝒫_{2}(ℝ^{d})), i = 0, 1, which converge strongly to u^{i}, i = 0, 1, such that

\underline{lim}_{n→∞} d((μ_{n}, u_{n}^{0})(μ_{n}, u_{n}^{1})) < min \{\frac{a}{8}, 2b\} .

In order to do so, find, for every n ∈ ℕ, a family of intervals I_{i}^{n} ⊂ ℝ, i = 1, …, n such that

I_{i}^{n} ∩ I_{j}^{n} = ∅, ∀n ∈ ℕ, 1 ≤ i ≤ n,
lim_{n→∞} sup_{i} H^{1}(I_{i}^{n}) = 0,
lim_{n→∞} I^{n} = [0, a],

where I^{n} := ∪_{i=1}^{n}I_{i}^{n} and the limit is intended w.r.t. the Hausdorff distance on compact sets. Now define A_{n} := {−1} × I^{n} ⊂ ℝ^{2}, B^{n} := {1} × I^{n} ⊂ ℝ^{2} and

μ_{n} := \frac{1}{2H^{1}(I^{n})}H^{1}|_{A^{n}∪B^{n}}.

Finally, define u_{n}^{0} as

u_{n}^{0}(x) := \{ \begin{matrix}(0, b), & x ∈ A^{n}, \\ (0, −b), & x ∈ B^{n}, \end{matrix}

and u_{n}^{1}(x) := −u_{n}^{0}(x). It is clear from the definition that (μ_{n}) converges to μ^{0} = μ^{1} w.r.t. W and (u_{n}^{i}) to u^{i} w.r.t. the strong convergence of maps.

Corollary 6.55 below ensures that, for every n ∈ ℕ, there exists a regular curve t → μ_{n}(t) whose length is given by

(6.57)  L_{0}^{1}(μ_{n}(t)) = \frac{2}{H^{1}(I^{n})} ∑_{i=1}^{n} H^{1}(I_{i}^{n}) (1 + \frac{(H^{1}(I_{i}^{n}))}{3}) ,

such that

T_{n}(A^{n}) = B^{n}, (6.58a) T_{n}(B^{n}) = A^{n}, (6.58b)

where T_{n}(x) := T_{n}(0, 1, x) and T_{n}(t, s, x) are the flow maps of t → μ_{n}(t). Equations (6.58), examples 6.40 and 6.42 ensure that T_{0}^{1}(u_{n}^{0}) = u_{n}^{1}. Therefore we obtain the inequality

\begin{matrix}d((μ_{n}, u_{n}^{0}), (μ_{n}, u_{n}^{1})) ≤ & (L_{0}^{1}(μ_{n}(t))^{2} + ∥T_{0}^{1}(u_{n}^{0}) − u_{n}^{1}∥_{μ_{n}(1)})^{1/2} \\ = & L_{0}^{1}(μ_{n}(t))^{2} ≤ \frac{2}{H^{1}(I^{n})} ∑_{i=1}^{n} H^{1}(I_{i}^{n}) (1 + \frac{(H^{1}(I_{i}^{n}))}{3}) . \end{matrix}

Passing to the limit we get

\underline{lim}_{n→∞} d((μ_{n}, u_{n}^{0}), (μ_{n}, u_{n}^{1})) ≤ 2,

therefore choosing a > 16 and b > 1 we get

d^{∗}((μ^{0}, u^{0}), (μ^{1}, u^{1})) ≤ \underline{lim}_{n→∞} d((μ_{n}, u_{n}^{0}), (μ_{n}, u_{n}^{1})) ≤ 2 < d((μ^{0}, u^{0}), (μ^{1}, u^{1})),

as desired.

Observe that, with minor but tedious modifications, it is possible to define the measures μ^{i}, μ_{n}^{i}, i = 0, 1, to be absolutely continuous with C^{∞} density.

Proposition 6.53. With the same notation as above, suppose that for a certain regular curve t → μ_{t} we are in case 2: that is, suppose that T(A) = B and T(B) = A, where T(x) := T(0, 1, x) and the maps T(t, s, x) are the flow maps of t → μ_{t}. Then the length of the curve is at least a/8.

Proof. For any two points x_{A} ∈ A and x_{B} ∈ B define the curve [0, 1] ∋ t → f(x_{A}, x_{b}, t) ∈ S^{1} (S^{1} being the unit circle in ℝ^{2}) as

f(x_{A}, x_{B}, t) := \frac{T(0, t, x_{B}) − T(0, t, x_{A})}{|T(0, t, x_{B}) − T(0, t, x_{A})|}.

Being the maps x → T(0, t, x) homeomorphism, they are injective: therefore the one above is a good definition and the function t → f(x_{A}, x_{B}, t) is continuous. Now let P := (0, −1) and extend the function f to the interval [−1, 2] by defining

\begin{aligned}f(x_{A}, x_{B}, t) & := \frac{(t + 1)(x_{B} − x_{A}) − tP}{|(t + 1)(x_{B} − x_{A}) − tP |}, ∀x ∈ [−1, 0), \\ f(x_{A}, x_{B}, t) & := \frac{(2 − t)(x_{A} − x_{B}) + (t − 1)P}{|(2 − t)(x_{A} − x_{B}) + (t − 1)P |}, ∀x ∈ (1, 2]. \end{aligned}

Given that the first component of x_{B}−x_{A} is strictly positive, the above equations define, for each x_{A}, x_{B}, a continuous map from [−1, 2] to S^{1}. Since we have f(x_{A}, x_{B}, −1) = P = f(x_{A}, x_{B}, 2), f is a closed loop, and therefore defines an element of the omothopy group of S^{1}. Now observe that f varies continuously in x_{A} and x_{B}, so the map which associates to (x_{A}, x_{B}) the element of the omothopy group is continuous. Since the omothopy group of S^{1} is discrete, this map has to be constant.

As a consequence we have that (at least) one of the following two must be true: • for every x_{A} ∈ A, x_{B} ∈ B the point P is in the image of

[0, 1] ∋ t → f(x_{A}, x_{B}, t),

• for every x_{A} ∈ A, x_{B} ∈ B the point −P is in the image of

[0, 1] ∋ t → f(x_{A}, x_{B}, t).

Suppose the first case is true (the other being similar). Define B′ := {1}×[3a/4, a] ⊂ B and A′ := {−1} × [0, a/4] ⊂ A and observe that for every x_{B} ∈ B′ and x_{A} ∈ A′ the second coordinate of x_{B} − x_{A} is at least a/2, while our assumption implies that for some t ∈ [0, 1] the second coordinate of T(0, t, x_{B}) − T(0, t, x_{A}) is negative. Therefore, defining the curves [0, 1] ∋ t → γ_{x}(t) := T(0, t, x) for every x ∈ A ∪ B, we have that L_{0}^{1}(γ_{x_{A}}(t)) + L_{0}^{1}(γ_{x_{B}}(t)) ≥ a/2. The conclusion follows by the chain of inequalities:

\begin{matrix}∫_{0}^{1} ∥v_{t}∥_{μ_{t}}dt = & ∫_{0}^{1} (∫_{ℝ^{2}} |v_{t}(x)|^{2}dμ_{t}(x))^{1/2} dt ≥ ∫_{0}^{1} ∫_{ℝ^{2}} |v_{t}(x)|dμ_{t}(x)dt \\ = & ∫_{ℝ^{2}} ∫_{0}^{1} |v_{t}(T(0, t, x))|dtdμ_{0}(x) = ∫_{ℝ^{2}} ∫_{0}^{1} |\frac{d}{dt}T(0, t, x)|dtdμ_{0}(x) \\ = & ∫_{ℝ^{2}} L_{0}^{1}(γ_{x})dμ_{0}(x) ≥ ∫_{A′∪B′} L_{0}^{1}(γ_{x})dμ_{0}(x) ≥ \frac{a}{8}. \end{matrix}

□

Proposition 6.54. With the same notation of the example above, there exists a regular curve t → μ_{t}, such that T(A) = B and T(B) = A whose length is given by 2(1 + a/3).

Proof. The idea is to move the mass as shown in the figure.

Step 1 Let A′ := {−1} × [0, a/3], B′ := {1} × [2/3a, a] and T ∈ L_{μ^{0}}^{2} be defined by

T (x_{1}, x_{2}) = \{ \begin{matrix}(x_{1}, x_{2}/3) , & (x_{1}, x_{2}) ∈ A, \\ (x_{1}, (2 + x_{2})/3) , & (x_{1}, x_{2}) ∈ B. \end{matrix}

Observe that the curve t → (Id + t(T − Id))_{#}μ^{0} is regular, that the support of the endpoint σ := T_{#}μ^{0} is A′ ∪ B′ and that its length is equal to ∥T − Id∥_{μ^{0}} = a/3

Step 2 Define S ∈ L_{σ}^{2} by

S(x_{1}, x_{2}) = \{ \begin{matrix}(1, x_{2}) , & (x_{1}, x_{2}) ∈ A′, \\ (−1, x_{2}) , & (x_{1}, x_{2}) ∈ B′. \end{matrix}

The curve t → (Id + t(S − Id))_{#}σ is regular and its length is 2. Observe that the endpoint of the curve is the transport of σ through the reflection on the vertical axis.

Step 3 We are in a situation symmetric to the one of the first step. Reversing the

construction given before we obtain a regular curve whose length is a/3. Given that a piecewise regular curve is regular, the proof is achieved. □

Corollary 6.55. With the same notation of the example above, there exists a regular curve t → μ_{n}(t) whose flow maps satisfy equation (6.58) and whose length is given by equation (6.57).

Proof. Fix n, then apply the construction given in the previous proposition to each one of the couples ({−1}×I_{i}^{n}, {1}×I_{i}^{n}) for i = 1, …, n. Given that during these movements the distance between the connected components of the support of μ_{t} is bounded from below by a positive constant uniformly on t, the resulting curve is regular. To conclude, observe that each couple ({−1} × I_{i}^{n}, {1} × I_{i}^{n}) gives a contribution to the total mass of H^{1}(I_{i}^{n})/H^{1}(I^{n}). □

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