Solving Poisson's equation for Wasserstein contractive Markov chains

TL;DR

This paper investigates solutions to Poisson's equation for Wasserstein contractive Markov chains, establishing existence, regularity, and maximal inequalities, with applications to MCMC methods.

Contribution

It provides new existence and regularity results for solutions to Poisson's equation under Wasserstein contraction and reversibility assumptions.

Findings

Solutions exist for Lipschitz functions under contraction.

Additional solutions for $L^p$ functions when the kernel is reversible.

Derived maximal inequalities for contractive Markov chains.

Abstract

We study Poisson's equation in the context of general state space Markov chains. For chains satisfying a contraction assumption w.r.t. a Wasserstein distance, we show that a solution exists for Lipschitz functions and investigate its regularity properties. If the kernel is additionally reversible we are also able to show that solutions for $L^p$ functions exist. Combining our findings with Doob's inequalities for martingales we derive maximal inequalities for contractive Markov chains. A number of examples is provided to demonstrate the applicability of our results, in particular in the context of Markov chain Monte Carlo methods.