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Nathael Gozlan

Publications and source records attributed to Nathael Gozlan.

At least 19 recordsLinked to original sources
arXiv2026

A Brenier-Strassen Theorem on CAT(kappa) Spaces

We extend the Brenier-Strassen theorem about projections in convex order to non-flat spaces with curvature bounded from above. Precisely, for probability measures $\mu$, $\nu$ of finite second moment on a complete separable CAT(0) space, we prove that $\mu$ admits a unique W 2 -projection \bar{\mu} to the set of probability measures dominated by $\nu$ in convex order. Moreover, the unique optimal coupling from $\mu$ to \bar{\mu} is induced by a 1-Lipschitz map, without any absolute-continuity assumption on $\mu$. Our proof identifies the projection problem with a weak optimal transport problem whose cost is the squared distance to the set of convex means. We also establish a localized version on CAT(kappa) spaces with kappa \geq 0, where the optimal map is 1/2-H{\"o}lder continuous. Finally, we give a Strassen-type characterization of one-step barycentric martingales on proper CAT(0) spaces.

math.FA↗
arXiv2024

Weak optimal transport with unnormalized kernels

We introduce a new variant of the weak optimal transport problem where mass is distributed from one space to the other through unnormalized kernels. We give sufficient conditions for primal attainment and prove a dual formula for this transport problem. We also obtain dual attainment conditions for some specific cost functions. As a byproduct we obtain a transport characterization of the stochastic order defined by convex positively 1-homogenous functions, in the spirit of Strassen theorem for convex domination.

math.FA↗
arXiv2024

Duality and Heat flow

We reveal the relation between the Legendre transform of convex functions and heat flow evolution, and how it applies to the functional Blaschke-Santalo inequality. We also describe local maximizers in this inequality.

math.FA↗
arXiv2024

Some obstructions to contraction theorems on the half-sphere

Caffarelli's contraction theorem states that probability measures with uniformly logconcave densities on R d can be realized as the image of a standard Gaussian measure by a globally Lipschitz transport map. We discuss some counterexamples and obstructions that prevent a similar result from holding on the half-sphere endowed with a uniform measure, answering a question of Beck and Jerison.

math.FA↗
arXiv2023

Transport-entropy forms of direct and Converseblaschke-Santal{ó} inequalities

We explore alternative functional or transport-entropy formulations of the Blaschke-Santal{ó} inequality and of its conjectured counterpart due to Mahler. In particular, we obtain new direct and reverse Blaschke-Santal{ó} inequalities for s-concave functions. We also obtain new sharp symmetrized transport-entropy inequalities for a large class of spherically invariant probability measures, including the uniform measure on the unit Euclidean sphere and generalized Cauchy and Barenblatt distributions. Finally, we show that the Mahler's conjecture is equivalent to some reinforced log-Sobolev type inequality on the sphere.

math.FA↗
arXiv2022

Log-Hessian and Deviation Bounds for Markov Semi-Groups, and Regularization Effect in $L^1$

It is well known that some important Markov semi-groups have a "regularization effect" -- as for example the hypercontractivity property of the noise operator on the Boolean hypercube or the Ornstein-Uhlenbeck semi-group on the real line, which applies to functions in $L^p$ for $p>1$. Talagrand had conjectured in 1989 that the noise operator on the Boolean hypercube has a further subtle regularization property for functions that are just integrable, but this conjecture remains open. Nonetheless, the Gaussian analogue of this conjecture was proven in recent years by Eldan-Lee and Lehec, by combining an inequality for the log-Hessian of the Ornstein-Uhlenbeck semi-group with a new deviation inequality for log-semi-convex functions under Gaussian measure. In this work, we explore the question of how much more general this phenomenon is. Specifically, our first goal is to explore the validity of both these ingredients for some diffusion semi-groups in $\mathbb{R}^n$, as well as for the $M/M/\infty$ queue on the non-negative integers and the Laguerre semi-groups on the positive real line. Our second goal is to prove a one-dimensional regularization effect for these settings, even in those cases where these ingredients are not valid.

math.PR↗
arXiv2021

Transport proofs of some functional inverse Santal{ó} inequalities

In this paper, we present a simple proof of a recent result of the second author which establishes that functional inverse-Santal{ó} inequalities follow from Entropy-Transport inequalities. Then, using transport arguments together with elementary correlation inequalities, we prove these sharp Entropy-Transport inequalities in dimension 1. We also revisit the proof of the functional inverse-Santal{ó} inequalities in the n dimensional unconditional case using these ideas.

math.FA↗
arXiv2020

Transport inequalities for random point measures

We derive transport-entropy inequalities for mixed binomial point processes, and for Poisson point processes. We show that when the finite intensity measure satisfies a Talagrand transport inequality, the law of the point process also satisfies a Talagrand type transport inequality. We also show that a Poisson point process (with arbitrary ${\sigma}$-finite intensity measure) always satisfies a universal transport-entropy inequality \`a la Marton. We explore the consequences of these inequalities in terms of concentration of measure and modified logarithmic Sobolev inequalities. In particular, our results allow one to extend a deviation inequality by Reitzner [31], originally proved for Poisson random measures with finite mass.

math.PR↗
arXiv2019

Transport Proofs Of Some Discrete Variants Of The Pr{é}Kopa-leindler Inequality

We give a transport proof of a discrete version of the displacement convexity of entropy on integers (Z), and get, as a consequence, two discrete forms of the Pr{é}kopa-Leindler Inequality : the Four Functions Theorem of Ahlswede and Daykin on the discrete hypercube [1] and a recent result on Z due to Klartag and Lehec [16].

math.PR↗
arXiv2019

A proof of the Caffarelli contraction theorem via entropic regularization

We give a new proof of the Caffarelli contraction theorem, which states that the Brenier optimal transport map sending the standard Gaussian measure onto a uniformly log-concave probability measure is Lipschitz. The proof combines a recent variational characterization of Lipschitz transport map by the second author and Juillet with a convexity property of optimizers in the dual formulation of the entropy-regularized optimal transport (or Schr{ö}dinger) problem.

math.PR↗
arXiv2018

On A Mixture Of Brenier and Strassen Theorems

We give a characterization of optimal transport plans for a variant of the usual quadratic transport cost introduced in [33]. Optimal plans are composition of a deterministic transport given by the gradient of a continuously differentiable convex function followed by a martingale coupling. We also establish some connections with Caffarelli's contraction theorem [14].

math.PR↗
arXiv2017

Deviation inequalities for convex functions motivated by the Talagrand conjecture

Motivated by Talagrand's conjecture on regularization properties of the natural semigroup on the Boolean hypercube, and in particular its continuous analogue involving regularization properties of the Ornstein-Uhlenbeck semigroup acting on in-tegrable functions, we explore deviation inequalities for log-semiconvex functions under Gaussian measure.

math.PR↗
arXiv2016

A variational approach to some transport inequalities

We relate transport-entropy inequalities to the study of critical points of functionals defined on the space of probability measures. This approach leads in particular to a new proof of a result by Otto and Villani [43] showing that the logarithmic Sobolev inequality implies Talagrand's transport inequality.

math.PR↗
arXiv2015

Kantorovich duality for general transport costs and applications

We introduce a general notion of transport cost that encompasses many costs used in the literature (including the classical one and weak transport costs introduced by Talagrand and Marton in the 90's), and prove a Kantorovich type duality theorem. As a by-product we obtain various applications in different directions: we give a short proof of a result by Strassen on the existence of a martingale with given marginals, we characterize the associated transport-entropy inequalities together with the log-Sobolev inequality restricted to convex/concave functions. Some explicit examples of discrete measures satisfying weak transport-entropy inequalities are also given.

math.PR↗
arXiv2015

Characterization of a class of weak transport-entropy inequalities on the line

We study an optimal weak transport cost related to the notion of convex order between probability measures. On the real line, we show that this weak transport cost is reached for a coupling that does not depend on the underlying cost function. As an application, we give a necessary and sufficient condition for weak transport-entropy inequalities in dimension one. In particular, we obtain a weak transport-entropy form of the convex Poincar{é} inequality in dimension one.

math.PR↗
arXiv2014

First order global asymptotics for confined particles with singular pair repulsion

We study a physical system of $N$ interacting particles in $\mathbb{R}^d$, $d\geq1$, subject to pair repulsion and confined by an external field. We establish a large deviations principle for their empirical distribution as $N$ tends to infinity. In the case of Riesz interaction, including Coulomb interaction in arbitrary dimension $d>2$, the rate function is strictly convex and admits a unique minimum, the equilibrium measure, characterized via its potential. It follows that almost surely, the empirical distribution of the particles tends to this equilibrium measure as $N$ tends to infinity. In the more specific case of Coulomb interaction in dimension $d>2$, and when the external field is a convex or increasing function of the radius, then the equilibrium measure is supported in a ring. With a quadratic external field, the equilibrium measure is uniform on a ball.

math.PR↗