On Transformations of Markov Chains and Poisson Boundary

TL;DR

This paper investigates how transformations of Markov chains via stopping times affect harmonic functions, providing conditions under which harmonic functions of the transformed chain relate to those of the original, with implications for Poisson boundaries.

Contribution

It establishes a sufficient condition on stopping times ensuring harmonic function spaces embed between original and transformed chains, extending previous work on random walks on groups.

Findings

Provides a sufficient condition for harmonic function embedding.

Extends results to positive unbounded harmonic functions.

Connects Markov chain transformations to Poisson boundary analysis.

Abstract

A discrete-time Markov chain can be transformed into a new Markov chain by looking at its states along iterations of an almost surely finite stopping time. By the optional stopping theorem, any bounded harmonic function with respect to the transition function of the original chain is harmonic with respect to the transition function of the transformed chain. The reverse inclusion is in general not true. Our main result provides a sufficient condition on the stopping time which guarantees that the space of bounded harmonic functions for the transformed chain embeds in the space of bounded harmonic sequences for the original chain. We also obtain a similar result on positive unbounded harmonic functions, under some additional conditions. Our work was motivated by and is analogous to Forghani-Kaimanovich, the well-studied case when the Markov chain is a random walk on a discrete group.