A variational approach to some transport inequalities

TL;DR

This paper introduces a variational framework linking transport-entropy inequalities to critical points of functionals on probability measures, providing a new proof that the logarithmic Sobolev inequality implies Talagrand's transport inequality.

Contribution

It presents a novel variational approach to connect transport inequalities with critical points of functionals, offering an alternative proof of a key implication.

Findings

Established a variational method relating transport-entropy inequalities to critical points

Provided a new proof that logarithmic Sobolev implies Talagrand's inequality

Enhanced understanding of the interplay between functional inequalities and probability measures

Abstract

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.