Limit behavior of the invariant measure for Langevin dynamics

TL;DR

This paper studies the limiting behavior of the invariant measure for Langevin dynamics with multiplicative noise as the noise amplitude approaches zero, showing convergence to a Gaussian measure under certain conditions.

Contribution

It proves the convergence of the scaled invariant measure to a Gaussian distribution and provides error estimates, extending understanding of Langevin dynamics in the small-noise limit.

Findings

Invariant measure converges to a Gaussian as noise vanishes

Convergence occurs in the p-Wasserstein distance for p in [1,2]

Error bounds for the approximation are established

Abstract

In this manuscript, we consider the Langevin dynamics on $\mathbb{R}^d$ with an overdamped vector field and driven by multiplicative Brownian noise of small amplitude $\sqrt{\epsilon}$, $\epsilon>0$. Under suitable assumptions on the vector field and the diffusion coefficient, 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 $p$-Wasserstein distance for $p\in [1,2]$ to a Gaussian measure with zero-mean vector and non-degenerate covariance matrix which solves a Lyapunov matrix equation. Moreover, the error term is estimated. We emphasize that generically no explicit formula for $\mu^{\epsilon}$ can be found.