Probabilistic approximation of a nonlinear parabolic equation occuring in rheology

In this paper, we are interested in a nonlinear parabolic evolution equation occuring in rheology. We give a probabilistic interpretation to this equation by associating a nonlinear martingale problem with it. We prove existence of a unique solution P to this martingale problem. For any t, the time-marginal at time t of P admits a density ρ(t, x) with respect to the Lebesgue measure and the function ρ is the unique weak solution of the evolution equation in a well-chosen energy space. Next, we introduce a simulable system of n interacting particles and prove that the empirical measure of this system converges to P as n tends to ∞. This propagation of chaos result ensures that the solution of the equation of interest can be approximated by a Monte-Carlo method. Last, we illustrate the convergence by some numerical experiments. AMS 2000 Mathematics Subject Classification: 60K35, 60F99, 65C35.