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

Full-text: Open access

Abstract

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.

Article information

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

Dates
First available in Project Euclid: 30 October 2008

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1225372947

Digital Object Identifier
doi:10.1214/07-AAP513

Mathematical Reviews number (MathSciNet)
MR2462546

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.aoap/1225372947.


Export citation

References

  • [1] Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C. and Scheffer, G. (2000). Sur les inégalités de Sobolev logarithmiques. Panoramas et Synthèses 10. Société Mathématique de France, Paris.
  • [2] Benachour, S., Roynette, B., Talay, D. and Vallois, P. (1998). Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos. Stochastic Process. Appl. 75 173–201.
  • [3] Benachour, S., Roynette, B. and Vallois, P. (1998). Nonlinear self-stabilizing processes. II. Convergence to invariant probability. Stochastic Process. Appl. 75 203–224.
  • [4] Brezis, H. (1983). Analyse Fonctionnelle. Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris.
  • [5] Carrillo, J. A., McCann, R. J. and Villani, C. (2003). Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates. Rev. Mat. Iberoamericana 19 971–1018.
  • [6] Cattiaux, P., Guillin, A. and Malrieu, F. (2008). Probabilistic approach for granular media equations in the non-uniformly convex case. Probab. Theory Related Fields 140 19–40.
  • [7] Fougères, P. (2005). Spectral gap for log-concave probability measures on the real line. In Séminaire de Probabilités XXXVIII. Lecture Notes in Mathematics 1857 95–123. Springer, Berlin.
  • [8] Jourdain, B. (2000). Diffusion processes associated with nonlinear evolution equations for signed measures. Methodol. Comput. Appl. Probab. 2 69–91.
  • [9] Jourdain, B. (2002). Probabilistic characteristics method for a one-dimensional inviscid scalar conservation law. Ann. Appl. Probab. 12 334–360.
  • [10] Jourdain, B. (2006). Probabilistic approximation via spatial derivation of some nonlinear parabolic evolution equations. In Monte Carlo and Quasi-Monte Carlo Methods 2004 197–216. Springer, Berlin.
  • [11] Ladyzenskaya, O. A., Solonnikov, V. A. and Ural’ceva, N. N. (1968). Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc., Providence, RI.
  • [12] Malrieu, F. (2001). Logarithmic Sobolev inequalities for some nonlinear PDE’s. Stochastic Process. Appl. 95 109–132.
  • [13] Malrieu, F. (2003). Convergence to equilibrium for granular media equations and their Euler schemes. Ann. Appl. Probab. 13 540–560.
  • [14] Pal, S. and Pitman, J. (2007). One-dimensional Brownian particle systems with rank dependent drifts. Available at http://arxiv.org/abs/0704.0957. Ann. Appl. Probab. To appear.