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Limite de champ moyen de systèmes de particules

François Bolley

Séminaire Équations aux dérivées partielles

On présente des résultats classiques et récents dans l’étude de la limite de champ moyen de systèmes de particules stochastiques en interaction. Ces derniers résultats visent à couvrir une plus grande variété de modèles et obtenir des estimations précises de la convergence et sont mises en lien avec le comportement en temps grand des systèmes considérés.

Quantitative concentration inequalities on sample path space for mean field interaction

François Bolley — 2010

ESAIM: Probability and Statistics

We consider the approximation of a mean field stochastic process by a large interacting particle system. We derive non-asymptotic large deviation bounds measuring the concentration of the empirical measure of the paths of the particles around the law of the process. The method is based on a coupling argument, strong integrability estimates on the paths in Hölder norm, and a general concentration result for the empirical measure of identically distributed independent paths.

Concentration of measure on product spaces with applications to Markov processes

Gordon BlowerFrançois Bolley — 2006

Studia Mathematica

For a stochastic process with state space some Polish space, this paper gives sufficient conditions on the initial and conditional distributions for the joint law to satisfy Gaussian concentration inequalities and transportation inequalities. In the case of the Euclidean space m , there are sufficient conditions for the joint law to satisfy a logarithmic Sobolev inequality. In several cases, the constants obtained are of optimal order of growth with respect to the number of random variables, or are...

Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation

François BolleyArnaud GuillinFlorent Malrieu — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a Vlasov-Fokker-Planck equation governing the evolution of the density of interacting and diffusive matter in the space of positions and velocities. We use a probabilistic interpretation to obtain convergence towards equilibrium in Wasserstein distance with an explicit exponential rate. We also prove a propagation of chaos property for an associated particle system, and give rates on the approximation of the solution by the particle system. Finally, a transportation inequality...

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