A new characterization of Talagrand's transport-entropy inequalities and applications

TL;DR

This paper establishes an equivalence between Talagrand's transport inequality and a restricted logarithmic Sobolev inequality, clarifying their relationship and demonstrating stability under bounded perturbations.

Contribution

It introduces a new characterization linking Talagrand's inequality to a restricted logarithmic Sobolev inequality and proves its stability under bounded perturbations.

Findings

Talagrand's inequality is equivalent to a restricted logarithmic Sobolev inequality

The equivalence clarifies the relationship between key functional inequalities

Talagrand's inequality remains stable under bounded perturbations

Abstract

We show that Talagrand's transport inequality is equivalent to a restricted logarithmic Sobolev inequality. This result clarifies the links between these two important functional inequalities. As an application, we give the first proof of the fact that Talagrand's inequality is stable under bounded perturbations.