# Inequalities

Tuesday, April 28, 2015 - 11:30am - 12:20pm

Giovanni Peccati (University of Luxembourg)

I will present a new set of functional inequalities involving the following four parameters associated with a given multidimensional distribution: the relative entropy, the relative Fisher information, the 2-Wasserstein distance, and the Stein discrepancy (which naturally appears in the well-known Stein's method for normal approximations). Our results improve the classical log-Sobolev and Talagrand's transport inequalities, and provide new key tools in order to deal with high-dimensional quantitative central limit theorems on a Gaussian space.

Tuesday, April 14, 2015 - 10:30am - 11:20am

Katalin Marton (Hungarian Academy of Sciences (MTA))

Abstract. The aim of this paper is to prove logarithmic Sobolev inequalities for

measures on discrete product spaces, by proving inequalities for an appropriate

Wasserstein-like distance. A logarithmic Sobolev inequality is, roughly speaking,

a contractivity property of relative entropy with respect to some Markov semigroup.

It is much easier to prove contractivity for a distance between measures, than for rela-

tive entropy, since for distances well known linear tools, like estimates through matrix

measures on discrete product spaces, by proving inequalities for an appropriate

Wasserstein-like distance. A logarithmic Sobolev inequality is, roughly speaking,

a contractivity property of relative entropy with respect to some Markov semigroup.

It is much easier to prove contractivity for a distance between measures, than for rela-

tive entropy, since for distances well known linear tools, like estimates through matrix

Friday, May 1, 2015 - 10:15am - 11:05am

Cyril Roberto (Université Paris Ouest)

We introduce a new notion of (weak) transport cost on the Euclidean space that, with the help of the Kantorovich duality presented in the lecture by Paul-Marie Samson on Thursday, will give rise to some applications: (1) a new very short proof of a result by Strassen, (2) a characterization of those probablity measures satisfying the log-Sobolev inequality restricted to convex functions, and (3) a characterization of those probability measures on the line satisfying some Talagrand type transport-entropy inequality.