PROPAGATION OF CHAOS AND POINCARE INEQUALITIES FOR A SYSTEM OF PARTICLES INTERACTING THROUGH THEIR CDF

• 成果类型: Article
• 署名作者: Jourdain, Benjamin; Malrieu, Florent
• 署名单位: Institut Polytechnique de Paris; Ecole Nationale des Ponts et Chaussees; Universite de Rennes
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/07-AAP513
• 发表日期: 2008
• 页码: 1706-1736
• 关键词: self-stabilizing processes; granular media equations; invariant probability; CONVERGENCE

摘要

In this paper, in the particular case of a concave flux function, we are interested in the long time behavior of the nonlinear process associated in [Methodol. Comput. Appl. Probab. 2 (2000) 69-91] to the one-dimensional viscous scalar conservation law. We also consider the particle system obtained by replacing the cumulative distribution function in the drift coefficient of this nonlinear process by the empirical cumulative distribution function. We first obtain a trajectorial propagation of chaos estimate which strengthens the weak convergence result obtained in [8] without any convexity assumption on the flux function. Then Poincare inequalities are used to get explicit estimates concerning the long time behavior of both the nonlinear process and the particle system.