Quantitative uniform in time chaos propagation for Boltzmann collision processes

摘要

This paper is devoted to the study of mean-field limit for systems ofindistinguables particles undergoing collision processes. As formulated by Kac\cite{Kac1956} this limit is based on the {\em chaos propagation}, and we (1)prove and quantify this property for Boltzmann collision processes withunbounded collision rates (hard spheres or long-range interactions), (2) proveand quantify this property \emph{uniformly in time}. This yields the firstchaos propagation result for the spatially homogeneous Boltzmann equation fortrue (without cut-off) Maxwell molecules whose "Master equation" sharessimilarities with the one of a L\'evy process and the first {\em quantitative}chaos propagation result for the spatially homogeneous Boltzmann equation forhard spheres (improvement of the %non-contructive convergence result ofSznitman \cite{S1}). Moreover our chaos propagation results are the firstuniform in time ones for Boltzmann collision processes (to our knowledge),which partly answers the important question raised by Kac of relating thelong-time behavior of a particle system with the one of its mean-field limit,and we provide as a surprising application a new proof of the well-known resultof gaussian limit of rescaled marginals of uniform measure on the$N$-dimensional sphere as $N$ goes to infinity (more applications will beprovided in a forthcoming work). Our results are based on a new method whichreduces the question of chaos propagation to the one of proving a purelyfunctional estimate on some generator operators ({\em consistency estimate})together with fine stability estimates on the flow of the limiting non-linearequation ({\em stability estimates}).