The Annals of Applied Probability

Convergence to equilibrium for granular media equations and their Euler schemes

Florent Malrieu

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Abstract

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

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

Dates
First available in Project Euclid: 18 April 2003

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1050689593

Digital Object Identifier
doi:10.1214/aoap/1050689593

Mathematical Reviews number (MathSciNet)
MR1970276

Zentralblatt MATH identifier
1031.60085

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

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

Citation

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. https://projecteuclid.org/euclid.aoap/1050689593.


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References

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