Abstract:

We consider the Langevin dynamics on R^d 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 ε^d/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 μ ^ε can be found.

2010 AMS Mathematics Subject Classification: Primary 60H10; Secondary 34D10, 37M25, 60F05, 49Q22.

Keywords and phrases: coupling, Gaussian distribution, invariant distribution, Langevin dynamics, Ornstein--Uhlenbeck process, perturbations of dynamical systems, Wasserstein distance.