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 $\sqrt{\epsilon}$, $\epsilon>0$. Under suitable assumptions diffusion coefficient, it is well-known that possesses a unique invariant probability measure $\mu^{\epsilon}$. As $\epsilon$ tends to zero, prove $\epsilon^{d/2} \mu^{\epsilon}(\sqrt{\epsilon}\mathrm{d} x)$ converges in $p$-Wasserstein distance for $p\in [1,2]$ Gaussian zero-mean non-degenerate covariance matrix which solves Lyapunov equation. Moreover, error term estimated. We emphasize generically no explicit formula $\mu^{\epsilon}$ can be found.