Open Access
May 2020 Around the entropic Talagrand inequality
Giovanni Conforti, Luigia Ripani
Bernoulli 26(2): 1431-1452 (May 2020). DOI: 10.3150/19-BEJ1163


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.


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Giovanni Conforti. Luigia Ripani. "Around the entropic Talagrand inequality." Bernoulli 26 (2) 1431 - 1452, May 2020.


Received: 1 September 2018; Revised: 1 September 2019; Published: May 2020
First available in Project Euclid: 31 January 2020

zbMATH: 07166569
MathSciNet: MR4058373
Digital Object Identifier: 10.3150/19-BEJ1163

Keywords: functional inequalities , large deviations , Optimal transport , semigroups

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

Vol.26 • No. 2 • May 2020
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