A stochastic particle method for some one-dimensional nonlinear p.d.e.

Abstract

We consider the one-dimensional nonlinear P.D.E. in the weak sense: ▪ When the initial condition is a probability on R , the solution U t is the distribution of the random variable X t where ( X t ) is a nonlinear stochastic process in the sense of McKean. Our purpose is to study a stochastic particle algorithm for the computation of the cumulative distribution function of U t . This method is based upon the moving of particles according to the law of a Markov chain approximating ( X t ), and the approximation of ( E b(x, X t) , t ≤ T) by means of empirical distributions. For a bounded Lipschitz function b, we prove the convergence of the method with a rate of convergence of order O(1/ N + A where N is the number of particles and Δ is the time step.