Nathael Gozlan

A characterization of dimension free concentration in terms of transportation inequalities.

The aim of this paper is to show that a probability measure concentrates independently of the dimension like a gaussian measure if and only if it verifies Talagrands

Hamilton Jacobi equations on metric spaces and transport entropy inequalities

\T_2

Poincaré inequalities and dimension free concentration of measure

transportation-cost inequality. This theorem permits us to give a new and very short proof of a result of Otto and Villani. Generalizations to other types of concentration are also considered. In particular, one shows that the Poincare inequality is equivalent to a certain form of dimension free exponential concentration. The proofs of these results rely on simple Large Deviations techniques.

A new characterization of Talagrand’s transport-entropy inequalities and applications

We prove an Hopf-Lax-Oleinik formula for the solutions of some Hamilton- Jacobi equations on a general metric space. As a first consequence, we show in full gener- ality that the log-Sobolev inequality is equivalent to an hypercontractivity property of the Hamilton-Jacobi semi-group. As a second consequence, we prove that Talagrands transport- entropy inequalities in metric space are characterized in terms of log-Sobolev inequalities restricted to the class of c-convex functions.

Characterization of Talagrand’s transport-entropy inequalities in metric spaces

In this paper, we consider Poincare inequalities for non euclidean metrics on

A variational approach to some transport inequalities

\mathbb{R}^d

The logarithmic Sobolev constant of the lamplighter

. These inequalities enable us to derive precise dimension free concentration inequalities for product measures. This technique is appropriate for a large scope of concentration rate: between exponential and gaussian and beyond. We give different equivalent functional forms of these Poincare type inequalities in terms of transportation-cost inequalities and infimum convolution inequalities. Workable sufficient conditions are given and a comparison is made with generalized Beckner-Latala-Oleszkiewicz inequalities.In this paper, we consider Poincaré inequalities for non euclidean metrics on R . These inequalities enable us to derive precise dimension free concentration inequalities for product measures. This technique is appropriate for a large scope of concentration rate: between exponential and gaussian and beyond. We give different equivalent functional forms of these Poincaré type inequalities in terms of transportation-cost inequalities and infimum convolution inequalities. Workable sufficient conditions are given and a comparison is made with generalized Beckner-Latala-Oleszkiewicz inequalities.

A large deviation approach to some transportation cost inequalities

We show that Talagrand’s transport inequality is equivalent to a restricted logarithmic Sobolev inequality. This result clarifies the links between these two important functional inequalities. As an application, we give the first proof of the fact that Talagrand’s inequality is stable under bounded perturbations.

Characterization of Talagrand's like transportation-cost inequalities on the real line

We give a characterization of transport-entropy inequalities in metric spaces. As an application we deduce that such inequalities are stable under bounded perturbation (Holley-Stroock perturbation Lemma).

Bounds on the deficit in the logarithmic Sobolev inequality

We relate transport-entropy inequalities to the study of critical points of functionals defined on the space of probability measures. This approach leads in particular to a new proof of a result by Otto and Villani [43] showing that the logarithmic Sobolev inequality implies Talagrands transport inequality.