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Contents of PMS, Vol. 42, Fasc. 1,
pages 143 - 162
DOI: 10.37190/0208-4147.00020
Published online 13.6.2022
 

Limit behavior of the invariant measure for Langevin~dynamics

G. Barrera

Abstract:

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/2με(√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.

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