Abstract not found

Abstract. In this paper, we give necessary and sufficient conditions for Talagrand’s like transportation cost inequalities on the real line. This brings a new wide class of examples of probability measures enjoying a dimension-free concentration of measure property. Another byproduct is the characterization of modified Log-Sobolev inequalities for Log-concave probability measures on R. 1.

In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner inequalities...). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi-group. In particular, we exhibit an example where Poincaré inequality can not be used for deriving entropic convergence whence weak logarithmic Sobolev inequality ensures the result.

In this paper, we consider Poincaré inequalities for non euclidean metrics on R d. 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.

This note is devoted to the proof of convex Sobolev (or generalized Poincaré) inequalities which interpolate between spectral gap (or Poincaré) inequalities and logarithmic Sobolev inequalities. We extend to the whole family of convex Sobolev inequalities results which have recently been obtained by Cattiaux [11] and Carlen and Loss [10] for logarithmic Sobolev inequalities. Under local conditions on the density of the measure with respect to a reference measure, we prove that spectral gap inequalities imply all convex Sobolev inequalities with constants which are uniformly bounded in the limit approaching the logarithmic Sobolev inequalities. We recover the case of the logarithmic Sobolev inequalities as a special case.

Logarithmic Sobolev inequality for log-concave measure from

Abstract. In this paper, we provide a characterization of a large class of transportation-cost inequalities in terms of exponential integrability of the cost function under the reference probability measure. Our results completely extend the previous works by Djellout, Guilin and Wu [8] and Bolley and Villani [3]. 1.

We develop in this paper an improvement of the method given by S. Bobkov and M. Ledoux in [BL00]. Using the Prékopa-Leindler inequality, we prove a modified logarithmic Sobolev inequality adapted for all measures on R n, with a strictly convex and super-linear potential. This inequality implies modified logarithmic Sobolev inequality, developed in [GGM05, GGM07], for all uniformly strictly convex potential as well as the Euclidean logarithmic Sobolev inequality. Résumé Dans cet article nous amélirons la méthode exposée par S. Bobkov et M. Ledoux dans [BL00]. En utilisant l’inégalité de Prékopa-Leindler, nous prouvons une inégalité de Sobolev logarithmique modifiée, adaptée à toutes les mesures sur R n possédant un potentiel strictement convexe et super-linéaire. Cette inégalité implique en particulier une inégalité de Sobolev logarithmique modifiée, développée dans [GGM05, GGM07], pour les mesures ayant un potentiel uniformément strictement convexe mais aussi une inégalité de Sobolev logarithmique de type euclidien. 1

Various properties of isoperimetric, functional, Transport-Entropy and concentration inequalities are studied on a Riemannian manifold equipped with a measure, whose generalized Ricci curvature is bounded from below. First, stability of these inequalities with respect to perturbation of the measure is obtained. The extent of the perturbation is measured using several different distances between perturbed and original measure, such as a one-sided L ∞ bound on the ratio between their densities, Wasserstein distances, and Kullback-Leibler divergence. In particular, an extension of the Holley–Stroock perturbation lemma for the log-Sobolev inequality is obtained. Second, the equivalence of Transport-Entropy inequalities with different cost functions is verified, by obtaining a reverse Jensen type inequality. In view of a recent result of Gozlan, this is used to obtain tensorization properties of concentration inequalities with respect to various product-metrics, and the tensorization result for isoperimetric inequalities of Barthe–Cattiaux–Roberto is easily recovered. Some further applications are also described. The main tool used is a previous precise result on the equivalence between concentration and isoperimetric inequalities in the described setting. 1

We find sufficient conditions for a probability measure µ to satisfy an inequality of the type Rd f 2 (