Around the entropic Talagrand inequality

TL;DR

This paper explores a generalized form of the Talagrand inequality involving entropic transportation costs, connecting it with Schrödinger bridges, hypercontractivity, and concentration of measure, and establishing its relation to classical inequalities.

Contribution

It introduces a new class of inequalities replacing Wasserstein distance with entropic costs, providing multiple characterizations and linking to existing functional inequalities.

Findings

Equivalent characterizations via reverse hypercontractivity and Hamilton-Jacobi-Bellman semigroup

Tensorization and relation to Logarithmic Sobolev inequalities

Implication of Talagrand inequality from the studied inequalities

Abstract

In this article we study generalization of the classical Talagrand transport-entropy inequality in which the Wasserstein distance is replaced by the entropic transportation cost. This class of inequalities has been introduced in the recent work [9], in connection with the study of Schr\"odinger bridges. We provide several equivalent characterizations in terms of reverse hypercontractivity for the heat semigroup, contractivity of the Hamilton-Jacobi-Bellman semigroup and dimension-free concentration of measure. Properties such as tensorization and relations to other functional inequalities are also investigated. In particular, we show that the inequalities studied in this article are implied by a Logarithmic Sobolev inequality and imply Talagrand inequality.