Présentation de la session.

Les inégalités fonctionnelles ont permis ces dernières années de mieux cerner le phénomène de concentration de la mesure, le comportement en temps long des processus de Markov, ou encore la courbure des espaces métriques mesurés, pour ne citer que quelques exemples. Elles jouent également un rôle important en mécanique statistique ou dans l’étude des propriétés asymptotiques des corps convexes. Cette session présentera quelques développements récents à la jonction entre probabilités, analyse et géométrie.

Les thèmes couverts seront les suivants : l’exposé de C. Léonard sera consacré aux liens entre courbure, processus de Markov et transport optimal ; E. Boissard parlera de la convergence des mesures d’occupations en distance de Wasserstein ; J. Lehec donnera dans son exposé de nouvelles preuves d’inspiration probabiliste d’inégalités fonctionnelles classiques ; G. Menz présentera quant à lui des résultats nouveaux sur les inégalités de Poincaré et de Sobolev Logarithmiques avec des applications en mécanique statistique.

Exposants :
- C. Léonard (Université Paris Ouest - Nanterre) 40 min
- E. Boissard (Université Toulouse 3) 20 min
- J. Lehec (Université Paris Dauphine) 20 min
- G. Menz (Max-Planck-Institut für Mathematik, Leipzig) 20 min.

Résumés des exposés :

- C. Léonard (U. Paris-Ouest) : Some transformations of Markov processes leading to curvature.

The first transformation we have in mind is a time-symmetric analogue of Doob’s h-transform which we call (f,g)-transform. It leads to the notion of "entropic interpolation" between two probability measures on a state space. It is a stochastic analogue of McCann’s interpolation which allows both recovering the basic results of the Bakry-Emery theory on a Riemannian manifold and extending it to a graph structure when considering continuous-time random walks, suggesting a natural definition of Ricci curvature on a graph.

The second transformation consists of slowing down to constant paths the (f,g)-transformed processes. In this limit, the entropic interpolation tends to some transport interpolation : a quadratic transport interpolation (McCann) in the Riemmannian setting and a metric transport interpolation in the graph setting.

We also consider some related functional inequalities : modified logarithmic Sobolev and Talagrand transport inequalities.

- E. Boissard (U. Toulouse 3) : Speed of convergence in the Wasserstein metrics for the L.L.N.

We discuss non-asymptotic bounds for the average speed of convergence of the empirical measure in the law of large numbers, in Wasserstein metrics. We also consider occupation measures of ergodic Markov chains. One motivation is the approximation of a probability measure by finitely supported measures (the quantization problem). It is found that rates for empirical or occupation measures match previously known optimal quantization rates in several cases. This is notably highlighted in the example of infinite-dimensional Gaussian measures.

- J. Lehec (U. Dauphine) : Representation formula for the entropy and functional inequalities.

We present a stochastic formula for the Gaussian relative entropy in the spirit of Borell’s formula for the Laplace transform. As an application, we shall give simple proofs of a number of functional inequalities.

- G. Menz (Max-Planck-Institut für Mathematik, Leipzig) : A two-scale proof of the Eyring-Kramers formula.

We consider a diffusion on a potential landscape which is given by a smooth Hamiltonian in the regime of small noise. We give a new proof of the Eyring-Kramers formula for the spectral gap of the associated generator of the diffusion. The proof is based on a refinement of the two-scale approach introduced by Grunewald, Otto, Villani, and Westdickenberg and of the mean-difference estimate introduced by Chafai and Malrieu. The Eyring-Kramers formula follows as a simple corollary from two main ingredients : The first one shows that the Gibbs measures restricted to a domain of attraction has a "good" Poincaré constant mimicking the fast convergence of the diffusion to metastable states. The second ingredient is the estimation of the mean-difference by a new weighted transportation distance. It contains the main contribution of the spectral gap, resulting from exponential long waiting times of jumps between metastable states of the diffusion. This new approach also allows to derive sharp estimates on the log-Sobolev constant.

Références.
∙ Ané, C. ; Blachère, S. ; Chafaï, D. ; Fougères, P. ; Gentil, I. ; Malrieu, F., Roberto, C. ; Scheffer, G. Sur les inégalités de Sobolev logarithmiques, Panoramas et Synthèses 10, SMF, Paris, 2000.
∙ Ledoux, M. The concentration of measure phenomenon, Mathematical Surveys and Monographs, 89. American Mathematical Society, Providence, RI, 2001.
∙ Villani, C. Optimal transport. Old and new. Grundlehren der Mathematischen Wissenschaften, 338. Springer-Verlag, Berlin, 2009.

Adresse organisateur :
Laboratoire d’Analyse et de Mathématiques Appliquées
UMR 8050
Université Paris-Est - Marne-la-Vallée
5, boulevard Descartes
Cité Descartes - Champs-sur-Marne
77454 Marne-la-Vallée Cedex 2