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

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.