The Annals of Applied Probability

A stochastic particle method with random weights for the computation of statistical solutions of McKean-Vlasov equations

Denis Talay and Olivier Vaillant

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Abstract

We are interested in statistical solutions of McKean-Vlasov-Fokker-Planck equations. An example of motivation is the Navier-Stokes equation for the vorticity of a two-dimensional incompressible fluid flow. We propose an original and efficient numerical method to compute moments of such solutions. It is a stochastic particle method with random weights. These weights are defined through nonparametric estimators of a regression function and convey the uncertainty on the initial condition of the considered equation. We prove an existence and uniqueness result for a class of nonlinear stochastic differential equations (SDEs), and we study the relationship between these nonlinear SDEs and statistical solutions of the corresponding McKean-Vlasov equations. This result forms the foundation of our stochastic particle method where we estimate the convergence rate in terms of the numerical parameters: the number of simulated particles and the time discretization step.

Article information

Source
Ann. Appl. Probab., Volume 13, Number 1 (2003), 140-180.

Dates
First available in Project Euclid: 16 January 2003

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1042765665

Digital Object Identifier
doi:10.1214/aoap/1042765665

Mathematical Reviews number (MathSciNet)
MR1951996

Zentralblatt MATH identifier
1026.60110

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 65C20: Models, numerical methods [See also 68U20] 65U05

Keywords
Stochastic particle methods statistical solution McKean-Vlasov equation

Citation

Talay, Denis; Vaillant, Olivier. A stochastic particle method with random weights for the computation of statistical solutions of McKean-Vlasov equations. Ann. Appl. Probab. 13 (2003), no. 1, 140--180. doi:10.1214/aoap/1042765665. https://projecteuclid.org/euclid.aoap/1042765665


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