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Shifted generalized Mehler semigroups on white noise functionals

Cig Un, Lee Mi Ra

Probability and Mathematical Statistics

2023

Vol. 43, Fasc. 2

185--205

We study shifted generalized Mehler semigroups on white noise functionals. We prove characterizations of invariant (white noise) distribution and the covariance property for shifted generalized Mehler semigroups. Also, we prove a Liouville type property of a shifted generalized Mehler semigroup or its infinitesimal generator.

Limit behavior of the invariant measure for Langevin~dynamics

Barrera Gerardo

Probability and Mathematical Statistics

2022

Vol. 42, Fasc. 1

143--162

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.