Limit behavior of the invariant measure for Langevin dynamics

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.

14 pages. Typos were corrected

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Cited by in corpus (1)

• Gradual convergence for Langevin dynamics on a degenerate potential