A stochastic particle method for McKean-Vlasov PDE's and the Burgers equation

From a probabilistic point of view, a McKean-Vlasov PDE is a limit equation for system of interacting particles and its weak solution mu(t) is a probability measure. We study a stochastic particle method for the computation of the cumulative distribution function of mu(t) based on, the simulation of the underlying particles systems. When the kernels of the PDE are Lipschitz and bounded functions, we prove that the convergence rate of the method is O(1/root N + root Delta t) for the L(1)(IR x Omega) norm of the error (N is the number of particles Delta t is the time step). We approximate the density of mu(t) by a smoothing of the discrete time empirical measure, The rate of convergence is of order O(epsilon(2) + 1/epsilon(1/root N + root Delta t)) (epsilon is the smoothing parameter), We extend the probabilistic interpretation of the nonlinear PDE and the stochastic particle method to a particular case of a discontinuous kernel corresponding to the Burgers equation. Nontrivial extensions of the methodology permit to obtain the convergence rate in O(1/root N + root Delta t) for the L(1)(IR x Omega) norm.