Limit behavior of the invariant measure for Langevin dynamics

We consider the Langevin dynamics on R d with an overdamped vector field and driven by multiplicative Brownian noise of small amplitude √ , > 0. Under suitable assumptions on the vector field and the diffusion coefficient, it is well-known that it has a unique invariant probability measure µ .We prove that as tends to zero, the probability measureconverges in the p-Wasserstein distance for p ∈ [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 µ can be found.