Limit behavior of the invariant measure for Langevin~dynamics

We consider the Langevin dynamics on Rd with an overdamped vector field and driven by multiplicative Brownian noise of small amplitude √ϵ, ϵ>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 ϵ . We prove that as ε tends to zero, the probability measure ϵd/2μ ϵ(√ϵdx) converges 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 μϵ can be found.