  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.
