## J. Dolbeault - B. Nazaret - G. SavarÃ©

# A new class of transport distances between measures

created by savare on 08 Mar 2008

modified on 12 Jan 2009

[

BibTeX]

*Published Paper*

**Inserted:** 8 mar 2008

**Last Updated:** 12 jan 2009

**Journal:** Calc. Var. Partial Differential Equations

**Volume:** 34

**Pages:** 193-231

**Year:** 2009

**Abstract:**

We introduce a new class of distances between
nonnegative Radon measures in $R^d$. They are modeled
on the dynamical characterization of the
Kantorovich-Rubinstein-Wasserstein
distances proposed by \textsc{Benamou-Brenier}
and provide a wide family interpolating
between the Wasserstein and the homogeneous
$W^{-1,p}$-Sobolev distances.

From the point of view of optimal transport theory,
these distances minimize a dynamical cost
to move a given initial distribution of mass
to a final configuration.
An important difference with the classical setting in mass transport
theory is that the cost not only depends on
the velocity of the moving particles but also on
the densities of the intermediate configurations with respect to
a given reference measure.

We study the topological and
geometric properties of these new distances,
comparing them with the notion of weak convergence of
measures and the well established
Kantorovich-Rubinstein-Wasserstein theory.
An example of possible applications
to the geometric theory of gradient flows is also given.

**Keywords:**
Optimal transport, Gradient flows, continuity equation

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