On the inclusion of bounded harmonic functions of random walks

TL;DR

This paper explores conditions under which bounded harmonic functions of one probability measure on a group are contained in those of another, using asymptotic commutativity and martingale techniques, with applications to the Choquet-Deny theorem.

Contribution

It introduces a new criterion based on asymptotic commutativity for the inclusion of harmonic function spaces, extending results without topological assumptions and applying martingale methods.

Findings

Asymptotic commutativity guarantees harmonic function inclusion.

Martingale techniques replace ergodic-theoretic arguments.

Results apply to general Markov chains and characterize harmonic function inclusion.

Abstract

We investigate the conditions under which the space of bounded harmonic functions of a probability measure $\mu$ on a group $G$ is contained in that of another measure $\theta$. We establish that asymptotic commutativity, defined by the condition $\|\mu^{*t}*\theta - \theta*\mu^{*t}\|_{TV} \to 0$ as $t \to \infty$, is sufficient to guarantee the inclusion $H^\infty(G, \mu) \subseteq H^\infty(G, \theta)$, provided $\theta$ is absolutely continuous with respect to a convex combination of convolution powers of $\mu$. By employing martingale convergence techniques rather than ergodic-theoretic arguments, we demonstrate that this result holds without topological assumptions on $G$ (such as local compactness) and extends to general Markov chains. Furthermore, utilizing hitting models for the Poisson boundary, we characterise the inclusion $H^\infty(G, \mu) \subseteq H^\infty(G, \theta)$ as equivalent to the asymptotic invariance of $\theta$ under $\mu$ in the weak* topology. We apply these results to provide a probabilistic proof of the Choquet-Deny theorem for nilpotent groups, among other applications.