A note on the $\mathcal{W}_2$-convergence rate of the empirical measure of an ergodic $\mathbb{R}^d$-valued diffusion

Abstract

In this note, we consider a Stochastic Differential Equation under a strong confluence and Lipschitz continuity assumption of the coefficients. For the unique stationary solution, we study the rate of convergence of its empirical measure toward the invariant probability measure. We provide rate for the Wasserstein distance in the mean quadratic and almost sure sense.