From the backward Kolmogorov PDE on the Wasserstein space to propagation of chaos for McKean-Vlasov SDEs

Abstract

This article is a continuation of our first work [6]. We here establish some new quantitative estimates for propagation of chaos of non-linear stochastic differential equations in the sense of McKean-Vlasov. We obtain explicit error estimates, at the level of the trajectories, at the level of the semi-group and at the level of the densities, for the mean-field approximation by systems of interacting particles under mild regularity assumptions on the coefficients. A first order expansion for the difference between the densities of one particle and its mean-field limit is also established. Our analysis relies on the well-posedness of classical solutions to the backward Kolmogorov partial differential equations defined on the strip [0,T]×Rd×P2(Rd), P2(Rd) being the Wasserstein space, that is, the space of probability measures on Rd with a finite second-order moment and also on the existence and uniqueness of a fundamental solution for the related parabolic linear operator here stated on [0,T]×P2(Rd).