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

Benjamin Jourdain (1) , Florent Malrieu (2)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique
2 IRMAR - Institut de Recherche Mathématique de Rennes

In the particular case of a concave flux function, we are interested in the long time behaviour of the nonlinear process associated to the one-dimensional viscous scalar conservation law. We also consider the particle system obtained by remplacing the cumulative distribution function in the drift coefficient of this nonlinear process by the empirical cdf. We first obtain trajectorial propagation of chaos result. Then, Poincaré inequalities are used to get explicit estimates concerning the long time behaviour of both the nonlinear process and the particle system.

Mots clés: viscous scalar conservation law; nonlinear process; interacting particle system; propagation of chaos; Poincaré inequality; long time behavior

Domaines: Probabilités [math.PR]

Benjamin Jourdain, Florent Malrieu. Propagation of chaos and Poincaré inequalities for a system of particles interacting through their cdf. The Annals of Applied Probability, 2008, 18 (5), pp.1706-1736. ⟨10.1214/07-AAP513⟩. ⟨hal-00127988v2⟩