Limit behavior of the invariant measure for Langevin dynamics

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 $\sqrt{ϵ}$, $ϵ>0$. Under suitable assumptions on the vector field and the diffusion coefficient, it is well-known that it possesses a unique invariant probability measure ${\mu }^{ϵ}$. As $ϵ$ tends to zero, we prove that the probability measure $\epsilon^{d/2} \mu^{\epsilon}(\sqrt{\epsilon}$\mathrm{d}$ x)$convergesinthe$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 ${\mu }^{ϵ}$ can be found.