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Abstract. µ being a nonnegative measure satisfying some Log-Sobolev inequality, we give conditions on F for the Boltzmann measure ν = e −2F µ to also satisfy some Log-Sobolev inequality. This paper improves and completes the final section in [6]. A general sufficient condition and a general necessary condition are given and examples are explicitly studied. Résumé. µ étant une mesure positive satisfaisant une inégalité de Sobolev logarithmique, nous donnons des conditions sur F pour que la mesure de Boltzmann ν = e −2F µ satisfasse également une telle inégalité (améliorant et complétant ainsi la dernière partie de [6]). Les conditions obtenues sont illustrées par des exemples.

We study in details the isoperimetric profile of product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate half-spaces are approximate solutions of the isoperimetric problem. 1

If a random variable is not exponentially integrable, it is known that no concentration inequality holds for an infinite sequence of independent copies. Under mild conditions, we establish concentration inequalities for finite sequences of n independent copies, with good dependence in n. 1

Abstract. We study the relationship between two classical approaches for quantitative ergodic properties: the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincaré type). We show that they can be linked through new inequalities (Lyapunov-Poincaré inequalities). Explicit examples for diffusion processes are studied, improving some results in the literature. The example of the kinetic Fokker-Planck equation recently studied by Hérau-Nier, Helffer-Nier and Villani is in particular discussed in the final section.

Large deviations and isoperimetric inequalities are considered for probability distributions, satisfying convexity conditions of the Brunn-Minkowski-type.

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Abstract. In this paper we derive non asymptotic deviation bounds for ∣∣ ∣ t

Abstract. This paper presents different approaches, based on functional inequalities, to study the speed of convergence in total variation distance of ergodic diffusion processes with initial law satisfying a given integrability condition. To this end, we give a general upper bound “à la Pinsker ” enabling us to study our problem firstly via usual functional inequalities (Poincaré inequality, weak Poincaré,...) and truncation procedure, and secondly through the introduction of new functional inequalities Iψ. These Iψ-inequalities are characterized through measure-capacity conditions and F-Sobolev inequalities. A direct study of the decay of Hellinger distance is also proposed. Finally we show how a dynamic approach based on reversing the role of the semi-group and the invariant measure can lead to interesting bounds. Résumé. Nous étudions ici la vitesse de convergence, pour la distance en variation totale, de diffusions ergodiques dont la loi initiale satisfait une intégrabilité donnée. Nous présentons différentes approches basées sur l’utilisation d’inégalités fonctionnelles. La première étape consiste à donner une borne générale à la Pinsker. Cette borne permet alors d’utiliser, en les combinant à une procedure de troncature, des inégalités usuelles (telles Poincaré ou Poincaré faibles,...). Dans un deuxième temps nous introduisons de nouvelles inégalités appelées Iψ que nous caractérisons à l’aide de condition de type capacité-mesure et d’inégalités de type F-Sobolev. Une étude directe de la distance de Hellinger est également proposée. Pour conclure, une approche dynamique basée sur le renversement du rôle du semigroupe de diffusion et de la mesure invariante permet d’obtenir de nouvelles bornes intéressantes. Key words: total variation, diffusion processes, speed of convergence, Poincaré inequality, logarithmic Sobolev inequality, F-Sobolev inequality. MSC 2000: 26D10, 60E15.

Using measure-capacity inequalities we study new functional inequalities, namely Lq-Poincaré inequalities Varµ(f q) 1/q ∫