Limit behavior of the invariant measure for Langevin dynamics

Abstract

In this article, we consider the Langevin dynamics on $\mathbb{R}^d$ with an overdamped vector field and driven by Brownian motion of small amplitude $\sqrt{\epsilon}$, $\epsilon>0$. Under suitable conditions on the vector field, it is well-known that it possesses a unique invariant probability measure $\mu^{\epsilon}$. As $\epsilon$ tends to zero, we prove that the probability measure $\epsilon^{d/2} \mu^{\epsilon}(\sqrt{\epsilon}\mathrm{d}x)$ converges in the $2$-Wasserstein distance to a Gaussian measure with zero-mean vector and non-degenerate covariance matrix which solves a Lyapunov matrix equation. We emphasize that generically no explicit formula for $\mu^{\epsilon}$ can be found.