CVGMT Papershttp://cvgmt.sns.it/papers/en-usSat, 24 Jun 2017 02:06:35 -0000Domains in metric measure spaces with boundary of positive mean curvature, and the Dirichlet problem for functions of least gradienthttp://cvgmt.sns.it/paper/3500/P. Lahti, L. Malý, N. Shanmugalingam, G. Speight.
<p>We study the geometry of domains in complete metric measure spaces equipped with a doubling measure
supporting a $1$-Poincar\'e inequality. We propose a notion of \emph{domain with boundary of positive
mean curvature} and prove that, for such domains, there is always a solution to the Dirichlet problem for
least gradients with continuous boundary data. Here \emph{least gradient} is defined as minimizing total
variation (in the sense of BV functions) and boundary conditions are satisfied in the sense that the
\emph{boundary trace} of the solution exists and agrees with the given boundary data. This
extends the result of Sternberg, Williams and Ziemer to the non-smooth setting. Via counterexamples
we also show that uniqueness of solutions and existence of \emph{continuous} solutions can fail, even
in the weighted Euclidean setting with Lipschitz weights.</p>
http://cvgmt.sns.it/paper/3500/Some remarks on boundary operators of Bessel extensionshttp://cvgmt.sns.it/paper/3499/J. Goodman, D. Spector.
<p>In this paper we study some boundary operators of a class of Bessel-type Littlewood-Paley extensions whose prototype is
\[
\Delta_x u(x,y) +\frac{1-2s}{y} \frac{\partial u}{\partial y}(x,y)+\frac{\partial^2 u}{\partial y^2}(x,y)=0 \text{ for }x\in\mathbb{R}^d, y>0,
\\
u(x,0)=f(x) \text{ for }x\in\mathbb{R}^d.
\]
In particular, we show that with a logarithmic scaling one can capture the failure of analyticity of these extensions in the limiting cases $s=k \in \mathbb{N}$.</p>
http://cvgmt.sns.it/paper/3499/Phase field approach to optimal packing problems and related Cheeger clustershttp://cvgmt.sns.it/paper/3498/B. Bogosel, D. Bucur, I. Fragalà.
<p> In a fixed domain of $\R ^N$ we study the asymptotic behaviour of optimal clusters associated to $\alpha$-Cheeger constants and natural energies like the sum or maximum: we prove that, as the parameter $\alpha$ converges to the ``critical" value $\Big (\frac{N-1}{N}\Big ) _+$, optimal Cheeger clusters converge to solutions of different packing problems for balls, depending on the energy under consideration. As well, we propose an efficient phase field approach based on a multiphase Gamma convergence result of Modica-Mortola type, in order to compute $\alpha$-Cheeger constants, optimal clusters and, as a consequence of the asymptotic result, optimal packings. Numerical experiments are carried over in two and three space dimensions.</p>
http://cvgmt.sns.it/paper/3498/Linearisation of multiwell energieshttp://cvgmt.sns.it/paper/3497/R. Alicandro, G. Dal Maso, G. Lazzaroni, M. Palombaro.
<p>Linear elasticity can be rigorously derived from finite elasticity under the assumption of small loadings in terms of Gamma-convergence. This was first done in the case of one-well energies with super-quadratic growth and later generalised to different settings, in particular to the case of multi-well energies where the distance between the wells is very small (comparable to the size of the load). In this paper we study the case when the distance between the wells is independent of the size of the load. In this context linear elasticity can be derived by adding to the multi-well energy a singular higher order term which penalises jumps from one well to another. The size of the singular term has to satisfy certain scaling assumptions whose optimality is shown in most of the cases. Finally, the derivation of linear elasticty from a two-well discrete model is provided, showing that the role of the singular perturbation term is played in this setting by interactions beyond nearest neighbours.
</p>
http://cvgmt.sns.it/paper/3497/Optimal estimates for the triple junction function and other surprising aspects of the area functionalhttp://cvgmt.sns.it/paper/3496/R. Scala.
<p>We consider the relaxed area functional for vector valued maps and its exact value on the triple junction function $u:B_1(O)\rightarrow\R^2$, a specific function which represents the first example of map whose graph area shows nonlocal effects. This is a map taking only three different values $\alpha,\beta,\gamma\in \R^2$ in three equal circular sectors of the unit radius ball $B_1(O)$. We prove a conjecture due to G. Bellettini and M. Paolini asserting that the recovery sequence provided in \cite{BP} (and the corresponding upper bound for the relaxed area functional of the map $u$) is optimal. At the same time, we show by means of a counterexample that such construction is not optimal if we consider different domains than $B_1(O)$, which still contain the same discontinuity set of $u$ in $B_1(O)$. Such domains are obtained from $B_1(O)$ erasing part of interior of the sectors where $u$ is constant.</p>
http://cvgmt.sns.it/paper/3496/Ollivier-Ricci idleness functions of graphshttp://cvgmt.sns.it/paper/3495/D. Bourne, D. Cushing, S. Liu, F. Münch, N. Peyerimhoff.
<p>We study the Ollivier-Ricci curvature of graphs as a function of the chosen idleness. We show that this idleness function is concave and piecewise linear with at most 3 linear parts, with at most 2 linear parts in the case of a regular graph. We then apply our result to show that the idleness function of the Cartesian product of two regular graphs is completely determined by the idleness functions of the factors.</p>
http://cvgmt.sns.it/paper/3495/Perturbations of Minimizing Movements and Curves of Maximal Slopehttp://cvgmt.sns.it/paper/3494/A. Tribuzio.
<p>We will perturbe the minimization algorithm of a functional $\phi$ in a metric space (S; d), introduced in the book "Gradient Flows in Metric Spaces and in the Space of Probability Measure" by L. Ambrosio, N.Gigli and G. Savarè, and prove that its minimizing movements are curves of maximal slope for $\phi$ with a perturbed velocity. We also show some cases in which the perturbed minimizing movements can escape from potential wells.</p>
<p>The topics discussed in this paper are a reworking of a part of my Master thesis "Perturbazioni di movimenti minimizzanti e curve di massima pendenza" through which I obtain a Master Degree in "Matematica Pura e Applicata" at the university of Rome "Tor Vergata". For this work therefore I would like to thank the Professor Andrea Braides, my thesis advisor, for his help and willingness that were fundamental for the success of this work.</p>
http://cvgmt.sns.it/paper/3494/Functions of Bounded Variation on the Classical Wiener Space and an
Extended Ocone-Karatzas Formulahttp://cvgmt.sns.it/paper/3493/M. Pratelli, D. Trevisan.
<p> We prove an extension of the Ocone-Karatzas integral representation, valid
for all $BV$ functions on the classical Wiener space. We establish also an
elementary chain rule formula and combine the two results to compute explicit
integral representations for some classes of $BV$ composite random variables.
</p>
http://cvgmt.sns.it/paper/3493/BV-regularity for the Malliavin Derivative of the Maximum of the Wiener
Processhttp://cvgmt.sns.it/paper/3491/D. Trevisan.
<p> We prove that, on the classical Wiener space, the random variable $\sup_{0\le
t \le T} W_t$ admits a measure as second Malliavin derivative, whose total
variation measure is finite and singular w.r.t.\ the Wiener measure.
</p>
http://cvgmt.sns.it/paper/3491/BV-capacities on Wiener Spaces and Regularity of the Maximum of the
Wiener Processhttp://cvgmt.sns.it/paper/3492/D. Trevisan.
<p> We define a capacity C on abstract Wiener spaces and prove that, for any u
with bounded variation, the total variation measure <br>Du<br> is absolutely
continuous with respect to C: this enables us to extend the usual rules of
calculus in many cases dealing with BV functions. As an application, we show
that, on the classical Wiener space, the random variable sup<sub>{0\leqt\leqT}</sub> W<sub>t</sub>
admits a measure as second derivative, whose total variation measure is
singular w.r.t. the Wiener measure.
</p>
http://cvgmt.sns.it/paper/3492/Uncertainty inequalities on groups and homogeneous spaces via
isoperimetric inequalitieshttp://cvgmt.sns.it/paper/3489/Gian Maria Dall'Ara, D. Trevisan.
<p> We prove a family of $L^p$ uncertainty inequalities on fairly general groups
and homogeneous spaces, both in the smooth and in the discrete setting. The
crucial point is the proof of the $L^1$ endpoint, which is derived from a
general weak isoperimetric inequality.
</p>
http://cvgmt.sns.it/paper/3489/Zero noise limits using local timeshttp://cvgmt.sns.it/paper/3490/D. Trevisan.
<p> We consider a well-known family of SDEs with irregular drifts and the
correspondent zero noise limits. Using (mollified) local times, we show which
trajectories are selected. The approach is completely probabilistic and relies
on elementary stochastic calculus only.
</p>
http://cvgmt.sns.it/paper/3490/A short proof of Stein's universal multiplier theoremhttp://cvgmt.sns.it/paper/3487/D. Trevisan.
<p> We give a short proof of Stein's universal multiplier theorem, purely by
probabilistic methods, thus avoiding any use of harmonic analysis techniques
(complex interpolation or transference methods).
</p>
http://cvgmt.sns.it/paper/3487/Lagrangian flows driven by $BV$ fields in Wiener spaceshttp://cvgmt.sns.it/paper/3488/D. Trevisan.
<p> We establish the renormalization property for essentially bounded solutions
of the continuity equation associated to $BV$ fields in Wiener spaces, with
values in the associated Cameron-Martin space; thus obtaining, by standard
arguments, new uniqueness and stability results for correspondent Lagrangian
$L^\infty$-flows. An example related to Neumann elliptic problems is also
discussed.
</p>
http://cvgmt.sns.it/paper/3488/Functional Cramer-Rao bounds and Stein estimators in Sobolev spaces, for
Brownian motion and Cox processeshttp://cvgmt.sns.it/paper/3485/E. Musta, M. Pratelli, D. Trevisan.
<p> We investigate the problems of drift estimation for a shifted Brownian motion
and intensity estimation for a Cox process on a finite interval $[0,T]$, when
the risk is given by the energy functional associated to some fractional
Sobolev space $H^1_0\subset W^{\alpha,2}\subset L^2$. In both situations,
Cramer-Rao lower bounds are obtained, entailing in particular that no unbiased
estimators with finite risk in $H^1_0$ exist. By Malliavin calculus techniques,
we also study super-efficient Stein type estimators (in the Gaussian case).
</p>
http://cvgmt.sns.it/paper/3485/Well-posedness of Multidimensional Diffusion Processes with Weakly
Differentiable Coefficientshttp://cvgmt.sns.it/paper/3486/D. Trevisan.
<p> We investigate well-posedness for martingale solutions of stochastic
differential equations, under low regularity assumptions on their coefficients,
widely extending some results first obtained by A. Figalli. Our main results
are a very general equivalence between different descriptions for
multidimensional diffusion processes, such as Fokker-Planck equations and
martingale problems, under minimal regularity and integrability assumptions,
and new existence and uniqueness results for diffusions having weakly
differentiable coefficients, by means of energy estimates and commutator
inequalities. Our approach relies upon techniques recently developed, jointly
with L. Ambrosio, to address well-posedness for ordinary differential equations
in metric measure spaces: in particular, we employ in a systematic way new
representations and inequalities for commutators between smoothing operators
and diffusion generators.
</p>
http://cvgmt.sns.it/paper/3486/A particle system approach to cell-cell adhesion modelshttp://cvgmt.sns.it/paper/3483/M. Neklyudov, D. Trevisan.
<p> We investigate micro-to-macroscopic derivations in two models of living
cells, in presence to cell-cell adhesive interactions. We rigorously address
two PDE-based models, one featuring non-local terms and another purely local,
as a a result of a law of large numbers for stochastic particle systems, with
moderate interactions in the sense of K. Oelshchlaeger (1985).
</p>
http://cvgmt.sns.it/paper/3483/Passive states optimize the output of bosonic Gaussian quantum channelshttp://cvgmt.sns.it/paper/3484/Giacomo De Palma, V. Giovannetti, D. Trevisan.
<p> An ordering between the quantum states emerging from a single mode
gauge-covariant bosonic Gaussian channel is proven. Specifically, we show that
within the set of input density matrices with the same given spectrum, the
element passive with respect to the Fock basis (i.e. diagonal with decreasing
eigenvalues) produces an output which majorizes all the other outputs emerging
from the same set. When applied to pure input states, our finding includes as a
special case the result of A. Mari, et al., Nat. Comm. 5, 3826 (2014) which
implies that the output associated to the vacuum majorizes the others.
</p>
http://cvgmt.sns.it/paper/3484/One-mode quantum-limited Gaussian channels have Gaussian maximizershttp://cvgmt.sns.it/paper/3481/Giacomo De Palma, V. Giovannetti, D. Trevisan.
<p> We prove that Gaussian states saturate the p->q norms of the one-mode
quantum-limited attenuator and amplifier. The proof starts from the
majorization result of De Palma et al., IEEE Trans. Inf. Theory 62, 2895
(2016), and is based on a new logarithmic Sobolev inequality. Our result
extends to noncommutative probability the seminal theorem "Gaussian kernels
have only Gaussian maximizers" (Lieb, Invent. Math. 102, 179 (1990)), stating
that Gaussian operators saturate the p->q norms of Gaussian integral kernels.
Our result also implies that the p->q norms of the thinning are saturated by
geometric probability distributions. Moreover, the multimode extension of our
result would imply the multiplicativity of the p->q norms of quantum-limited
Gaussian channels.
</p>
http://cvgmt.sns.it/paper/3481/Gaussian States Minimize the Output Entropy of the One-Mode Quantum
Attenuatorhttp://cvgmt.sns.it/paper/3482/Giacomo De Palma, V. Giovannetti, D. Trevisan.
<p> We prove that Gaussian thermal input states minimize the output von Neumann
entropy of the one-mode Gaussian quantum-limited attenuator for fixed input
entropy. The Gaussian quantum-limited attenuator models the attenuation of an
electromagnetic signal in the quantum regime. The Shannon entropy of an
attenuated real-valued classical signal is a simple function of the entropy of
the original signal. A striking consequence of energy quantization is that the
output von Neumann entropy of the quantum-limited attenuator is no more a
function of the input entropy alone. The proof starts from the majorization
result of De Palma et al., IEEE Trans. Inf. Theory 62, 2895 (2016), and is
based on a new isoperimetric inequality. Our result implies that geometric
input probability distributions minimize the output Shannon entropy of the
thinning for fixed input entropy. Moreover, our result opens the way to the
multimode generalization, that permits to determine both the triple trade-off
region of the Gaussian quantum-limited attenuator and the classical capacity
region of the Gaussian degraded quantum broadcast channel.
</p>
http://cvgmt.sns.it/paper/3482/