The Annals of Applied Probability

Propagation of chaos and Poincaré inequalities for a system of particles interacting through their cdf

Benjamin Jourdain and Florent Malrieu

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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 Poincaré inequalities are used to get explicit estimates concerning the long time behavior of both the nonlinear process and the particle system.

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Ann. Appl. Probab. Volume 18, Number 5 (2008), 1706-1736.

First available in Project Euclid: 30 October 2008

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Primary: 65C35: Stochastic particle methods [See also 82C80] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60E15: Inequalities; stochastic orderings 35K15: Initial value problems for second-order parabolic equations 46N30: Applications in probability theory and statistics

Viscous scalar conservation law nonlinear process particle system propagation of chaos Poincaré inequality long time behavior


Jourdain, Benjamin; Malrieu, Florent. Propagation of chaos and Poincaré inequalities for a system of particles interacting through their cdf. Ann. Appl. Probab. 18 (2008), no. 5, 1706--1736. doi:10.1214/07-AAP513.

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