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

Journal article
Authors S. Mischler
C. Mouhot
Bernt Wennberg
Published in Probability Theory and Related Fields
Volume 161
Issue 1-2
Pages 1-59
ISSN 0178-8051
Publication year 2015
Published at Department of Mathematical Sciences
Pages 1-59
Language en
Links dx.doi.org/10.1007/s00440-013-0542-...
Keywords Boltzmann equation , Drift-diffusion , Fluctuations , Granular gas , Inelastic collision , McKean-Vlasov equation , Mean field limit , Quantitative
Subject categories Mathematical Analysis, Probability Theory and Statistics

Abstract

This paper is devoted 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. © 2013 Springer-Verlag Berlin Heidelberg.

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