A new approach to quantitative propagation of chaos for drift, diffusion and jump processes

Stéphane Mischler (1) , Clément Mouhot (2) , Bernt Wennberg (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 DPMMS - Department of Pure Mathematics and Mathematical Statistics
3 Department of Mathematical Sciences

This paper is devoted the the study of the mean field limit for many-particle systems undergoing jump, drift or diffusion processes, as well as combinations of them. The main results are quantitative estimates on the decay of fluctuations around the deterministic limit and of correlations between particles, as the number of particles goes to infinity. To this end we introduce a general functional framework which reduces this question to the one of proving a purely functional estimate on some abstract generator operators (consistency estimate) together with fine stability estimates on the flow of the limiting nonlinear equation (stability estimates). Then we apply this method to a Boltzmann collision jump process (for Maxwell molecules), to a McKean-Vlasov drift-diffusion process and to an inelastic Boltzmann collision jump process with (stochastic) thermal bath. To our knowledge, our approach yields the first such quantitative results for a combination of jump and diffusion processes.

Mots clés: mean-field limit; quantitative; fluctuations; Boltzmann equation; McKean-Vlasov equation; drift-diffusion; inelastic collision; granular gas

Domaines: Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]

Stéphane Mischler, Clément Mouhot, Bernt Wennberg. A new approach to quantitative propagation of chaos for drift, diffusion and jump processes. Probability Theory and Related Fields, 2013, pp.online first. ⟨10.1007/s00440-013-0542-8⟩. ⟨hal-00559132v3⟩