Abstract not found

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.

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.

We consider the coarse-graining of a lattice system with continuous spin variable. In the first part, two abstract results are established: sufficient conditions for a logarithmic Sobolev inequality with constants independent of the dimension (Theorem 3) and sufficient conditions for convergence to the hydrodynamic limit (Theorem 8). In the second part, we use the abstract results to treat a specific example, namely the Kawasaki dynamics with Ginzburg–

We consider an N–site lattice system with continuous spin variables governed by a Ginzburg–Landau–type potential. Because we are interested in the Kawasaki dynamics, we work with the canonical ensemble in which the mean m is given. We prove a logarithmic Sobolev inequality (LSI) which is uniform in m and has the optimal scaling in the system size N. The method involves a two–scale “block–spin ” decomposition. Choosing sufficiently large blocks leads to convexification of the coarse–grained Hamiltonian; consequently, the Bakry–Emery principle implies a macroscopic LSI. On the other hand, the Holley–Stroock lemma implies a microscopic LSI as long as the block–spin size is bounded. We show that the macro – and microscopic LSI can be combined to yield a global LSI. The main ingredient in this final step is the Talagrand inequality.

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.

We prove in this paper an existence result for in nite-dimensional stationary weakly interactive Brownian diusions. The interaction is very general in the sense that it is not supposed to be regular, and it also could be non-Markovian, but it is small enough. Our method consists in using the characterization of such diusions as space-time Gibbs elds so that we construct them by space-time cluster expansions in the small coupling parameter.

Competitive or weak cooperative stochastic Lotka-Volterra systems conditioned to non-extinction

Quasi-stationarity distributions and diffusion models in population dynamics

Abstract. We show that the quadratic transportation cost inequality T2 is equivalent to both a Poincaré inequality and a strong form of the Gaussian concentration property. In particular if a logarithmic Sobolev inequality implies T2, we are able to give examples for which T2 holds but the logarithmic Sobolev inequality does not hold. This answers to a question left open by Otto and Villani [21] and Bobkov, Gentil and Ledoux [4], and furnishes (in a Riemannian setting) the analogue of the well known criterion by Bobkov and Götze for the linear transportation cost inequality T1 [5] (also see [12]). The main ingredient in the proof is a new family of inequalities, called modified quadratic transportation cost inequalities in the spirit of the modified logarithmic-Sobolev inequalities by Bobkov and Ledoux [6], that are shown to hold as soon as a Poincaré inequality is satisfied.