Around the entropic Talagrand inequality

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 (Probab. Theory Related Fields 174 (2019) 1-47), in connection with the study of Schrödinger 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.