PROPAGATION OF CHAOS AND FLUCTUATIONS FOR A MODERATE MODEL WITH SMOOTH INITIAL DATA

In this paper, we are interested in a stochastic differentialequation which is nonlinear in the following sense : both the diffusion and the drift coefficients depend locally on the density of the time marginal of the solution. When the law of the initial data has a smooth densit y with respect to Lebesgue measure, we prove existence and uniqueness for this equation. Under more restrictive assumptions on the density, we approximate the solution by a system of n moderately interacting di ffusion processes and obtain a trajectorial propagation of chaos resul t. Finally, we study the fluctuations associated with the convergence of the empirical measure of the system to the law of the solution of t he nonlinear equation. In this situation, the convergence rate is diff erent from root n. (C) Elsevier, Paris.