Limit behavior of the invariant measure for Langevin~dynamics

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/2}*μ*^{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.