Inequalities

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.

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

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.