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

TL;DR

This paper develops a general framework to quantitatively analyze the mean field limit for systems with jump, drift, and diffusion processes, providing new results for combined jump-diffusion systems.

Contribution

It introduces a novel functional approach that yields the first quantitative estimates for systems combining jump and diffusion dynamics.

Findings

Quantitative decay estimates for fluctuations around the mean field limit.

Application of the method to Boltzmann collision and McKean-Vlasov processes.

First quantitative results for systems with both jump and diffusion components.

Abstract

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.