The Annals of Applied Probability

Convergence to equilibrium for granular media equations and their Euler schemes

Florent Malrieu

Full-text: Open access


We introduce a new interacting particle system to investigate the behavior of the nonlinear, nonlocal diffusive equation already studied by Benachour et al. [3, 4]. We first prove an uniform (with respect to time) propagation of chaos. Then, we show that the solution of the nonlinear PDE converges exponentially fast to equilibrium recovering a result established by an other way by Carrillo, McCann and Vilanni [7]. At last we provide explicit and Gaussian confidence intervals for the convergence of an implicit Euler scheme to the stationary distribution of the nonlinear equation.

Article information

Ann. Appl. Probab. Volume 13, Number 2 (2003), 540-560.

First available in Project Euclid: 18 April 2003

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65C35: Stochastic particle methods [See also 82C80]
Secondary: 35K55: Nonlinear parabolic equations 65C05: Monte Carlo methods 82C22: Interacting particle systems [See also 60K35]

Interacting particle system propagation of chaos logarithmic Sobolev inequality nonlinear parabolic PDE concentration of measure phenomenon implicit Euler scheme


Malrieu, Florent. Convergence to equilibrium for granular media equations and their Euler schemes. Ann. Appl. Probab. 13 (2003), no. 2, 540--560. doi:10.1214/aoap/1050689593.

Export citation


  • ROBERTO, C. and SCHEFFER, G. (2000). Sur les inéqualités de Sobolev logarithmiques. Société Mathématique de France, Paris.
  • [2] BAKRY, D. (1997). On Sobolev and logarithmic Sobolev inequalities for Markov semigroups. In New Trends in Stochastic Analy sis (K. D. Elworthy, S. Kusuoka and I. Shigekawa, eds.) 43-75. World Scientific, River Edge, NJ.
  • [3] BENACHOUR, S., ROy NETTE, B., TALAY, D. and VALLOIS, P. (1998). Nonlinear selfstabilizing processes. I. Existence, invariant probability, propagation of chaos. Stochastic Process. Appl. 75 173-201.
  • [4] BENACHOUR, S., ROy NETTE, B. and VALLOIS, P. (1998). Nonlinear self-stabilizing processes II. Convergence to invariant probability. Stochastic Process. Appl. 75 203-224.
  • [5] BENEDETTO, D., CAGLIOTI, E. and PULVIRENTI, M. (1997). A kinetic equation for granular media. RAIRO Modél. Math. Anal. Numér. 31 615-641.
  • [6] BOBKOV, S., GENTIL, I. and LEDOUX, M. (2001). Hy percontractivity of Hamilton-Jacobi equations. J. Math. Pure Appl. 80 669-696.
  • [7] CARRILLO, J. A., MCCANN, R. J. and VILLANI, C. (2002). Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates. Rev. Mat. Iberoamericana. To appear.
  • [8] LEDOUX, M. (1999). Concentration of measure and logarithmic Sobolev inequalities. Séminaire de Probabilités XXXIII. Lecture Notes in Math. 1709 120-216. Springer, Berlin.
  • [9] OTTO, F. and VILLANI, C. (2000). Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 361-400.
  • [10] TALAY, T. (2002). Stochastic Hamiltonian dissipative sy stems: exponential convergence to the invariant measure, and discretisation by the implicit Euler schemes. Markov Process. Appl. 8 163-198.