Law of the Iterated Logarithm for Markov Semigroups with Exponential Mixing in the Wasserstein Distance

TL;DR

This paper proves the law of the iterated logarithm for a broad class of non-stationary Markov processes with exponential mixing in Wasserstein distance, extending classical results to infinite-dimensional stochastic systems.

Contribution

It establishes the law of the iterated logarithm for non-stationary Markov processes with exponential Wasserstein mixing, including applications to infinite-dimensional SDEs.

Findings

Law of the iterated logarithm proven for certain Markov processes

Applicable to infinite-dimensional stochastic differential equations

Demonstrates exponential mixing in Wasserstein distance

Abstract

In this paper, we establish the law of the iterated logarithm for a wide class of non-stationary, continuous-time Markov processes evolving on Polish spaces. Specifically, our result applies to certain additive functionals of processes governed by stochastically continuous Markov-Feller semigroups that exhibit exponential mixing and non-expansiveness in the Wasserstein distance, provided that a suitable moment condition involving the initial distribution is satisfied. Furthermore, we outline the application of this result to a Markov process arising as the solution of an infinite-dimensional stochastic differential equation with dissipative drift and additive noise.