We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a characterization of those probablity measures which satisfy these inequalities.

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We survey recent works on the connection between spectral gap and logarithmic Sobolev constants, and exponential integrability of Lipschitz functions. In particular, tools from measure concentration are used to describe bounds on the diameter of a (compact) Riemannian manifold and of Markov chains in terms of the first eigenvalue of the Laplacian and the logarithmic Sobolev constant. We examine similarly dimension free isoperimetric bounds using these parameters.

Abstract. We study the Langevin dynamics for the family of spherical p-spin disordered meanfield models of statistical physics. We prove that in the limit of system size N approaching infinity, the empirical state correlation and integrated response functions for these N-dimensional coupled diffusions converge almost surely and uniformly in time, to the non-random unique strong solution of a pair of explicit non-linear integro-differential equations, first introduced by Cugliandolo and Kurchan.

Brascamp–Lieb-type, weighted Poincaré-type and related analytic inequalities are studied for multidimensional Cauchy distributions and more general κ-concave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinitedimensional log-concave) Gaussian model, the weighted inequalities fully describe the measure concentration and large deviation properties of this family of measures. Cheeger-type isoperimetric inequalities are investigated similarly, giving rise to a common weight in the class of concave probability measures under consideration. 1. Introduction. The

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.

By using perturbation arguments, a sufficient condition is presented for the ultracontractivity of symmetric diffusion semigroups. As a consequence, a result suggested by D. Stroock is proved: let P t be generated by L = + rV with V = r ( > 0; > 2) on a complete connected Riemannian manifold M , where is the Riemannian distance function from a fixed point, then P t is ultracontractive provided the Ricci curvature is bounded below. Furthermore, kP t k1!1 exp[ 1 + 2 t =( 2) ] for some 1 ; 2 > 0: Next, it is shown that, for a di usion semigroup (not necessarily symmetric) with an invariant probability measure, if the curvature of its generator is bounded from below, then the ultracontractivity is equivalent to kP t exp[ 2 ]k 1 0: Especially, the above estimate of kP t k1!1 holds if L 2 c1 c2 for some c1 ; c2 > 0 and > 2: This estimate is sharp as is shown by examples given at the end of the paper. Corresponding results are proven for supe...