Dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp–Lieb inequalities

Abstract

In this work we consider dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp-Lieb inequalities. For this we use optimal transport methods and the Borell-Brascamp-Lieb inequality. These refinements can be written as a deficit in the classical inequalities. They have the right scale with respect to the dimension. They lead to sharpened concentration properties as well as refined contraction bounds, convergence to equilibrium and short time behavior for the laws of solutions to stochastic differential equations.

Concentration of information content for convex measures

“…The main result of this section is the following theorem which unifies and extends previous results of Borell [10], Bobkov and Madiman [5], and Fradelizi, Madiman and Wang [18] (see also Bolley-Gentil-Guillin [9]). A weaker log-concavity statement was also obtained by Nguyen [32].…”

“…In Section 2, we will show that exponential deviation of a functional follows from the log-concavity of the normalized Laplace transform of that functional. In particular, we study the log-concavity of the normalized moments of s-concave functions in Section 3, and present a sharp result that unifies and extends results of [10,5,18,9,32]. In Section 4, we examine a related monotonicity property of normalized moments, giving in particular an extension of a result of [21].…”

Convexity and regularity properties for entropic interpolations

“…✄ Remark 14 Contraction for the entropic cost, implies the analogue dimensional contraction for the quadratic Wasserstein distance along the heat flow [BGG16,BGGK16].…”

A sharp symmetrized form of Talagrand’s transport-entropy inequality for the Gaussian measure

“…where E is the set of functions for which equality holds, and d is some suitable distance on the space of functions or measures considered. This problem has been recently studied for several Gaussian functional inequalities, such as the isoperimetric problem [16,5], the logarithmic Sobolev inequality [12,17,8] and Talagrand's inequality [17,13]. In particular, [23] investigated applications of the moment map problem to such deficit estimates.…”