Quantitative chaos propagation estimates for jump processes

Together with my collaborator Stéphane Mischler we have uploaded this morning a new paper on the issue of mean-field limit. It provides a new systematic method for proving in a quantitative manner estimates of chaos propagation which measure fluctuations around the limit deterministic mean-field dynamics, and it is applied to collision jump processes as considered by Kac in the 50ies.

http://hal.archives-ouvertes.fr/hal-00447988/fr/

http://fr.arxiv.org/abs/1001.2994

In short we prove:
(1) Quantitative chaos propagation estimates (and therefore mean-field limit) for the spatially homogeneous Boltzmann equation for two physical interactions:
(2) True Maxwell molecules (long-range interaction, ressembling a Levy process)
(3) Hard spheres
(4) Finally these estimates are uniform in time (allowing to commute mean-field limit and time asymptotics).
This is based on a new method, formulated as the consistency/stability analysis of the convergence of a numerical scheme, whose main specific feature is an assumption of differential stability on the limit semigroup.

Further papers are in preparation. In particuler a paper applying the method to dissipative particule systems together with Bernt Wennberg should be available soon.

Abstract:

This paper is devoted to the study of mean-field limit for systems of indistinguables particles undergoing collision processes. As formulated by [Kac, 1956] this limit is based on the chaos propagation, and we (1) prove and quantify this property for Boltzmann collision processes with unbounded collision rates (hard spheres or long-range interactions), (2) prove and quantify this property \emph{uniformly in time}. This yields the first chaos propagation result for the spatially homogeneous Boltzmann equation for true (without cut-off) Maxwell molecules whose “Master equation” shares similarities with the one of a Lévy process and the first quantitative chaos propagation result for the spatially homogeneous Boltzmann equation for hard spheres (improvement of the convergence result of [Sznitman, 1984]. Moreover our chaos propagation results are the first uniform in time ones for Boltzmann collision processes (to our knowledge), which partly answers the important question raised by Kac of relating the long-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 result of gaussian limit of rescaled marginals of uniform measure on the N-dimensional sphere as N goes to infinity (more applications will be provided in a forthcoming work). Our results are based on a new method which reduces the question of chaos propagation to the one of proving a purely functional estimate on some generator operators (consistency estimate) together with fine stability estimates on the flow of the limiting non-linear equation (stability estimates).

One comment on “Quantitative chaos propagation estimates for jump processes

  1. Yong-Kum Cho says:

    Hi, I am a mathematician from Korea. My recent research interests include Boltzmann equations and I
    get to know you by your papers. During April 22-April 28, I plan to visit London and I wonder if you are still in Cambridge and if I have a chance to visit you if so.

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