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On the inclusion of bounded harmonic functions of random walks

Yair Hartman, Aranka Hrušková, Omer Segev

TL;DR

The paper addresses when spaces of bounded harmonic functions for two random-walk measures on a group satisfy $H^ fty(G,\mu)\subseteq H^ fty(G,\theta)$. It develops a martingale-based approach built on asymptotic commutativity: if $\theta\ll\sum_{n\ge0} 2^{-(n+1)}\mu^{*n}$ and $\|\mu^{*t}*\theta-\theta*\mu^{*t}\|_{TV}\to0$, then the inclusion holds, with equality when $\mu$ and $\theta$ commute and generate $G$. The authors extend the framework to general Markov chains, provide a Derriennic-type necessary-and-sufficient characterization via total variation and weak*-convergence on boundary spaces, and relate these conditions to hitting models of the Poisson boundary. They then derive several applications, including a probabilistic proof of the Choquet-Deny theorem for nilpotent groups and structural results about the center and product groups. Overall, the work bridges probabilistic martingale methods with boundary theory to obtain robust inclusions and equivalences for harmonic functions beyond locally compact second countable groups.

Abstract

We investigate the conditions under which the space of bounded harmonic functions of a probability measure $μ$ on a group $G$ is contained in that of another measure $θ$. We establish that asymptotic commutativity, defined by the condition $\|μ^{*t}*θ- θ*μ^{*t}\|_{TV} \to 0$ as $t \to \infty$, is sufficient to guarantee the inclusion $H^\infty(G, μ) \subseteq H^\infty(G, θ)$, provided $θ$ is absolutely continuous with respect to a convex combination of convolution powers of $μ$. 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 $H^\infty(G, μ) \subseteq H^\infty(G, θ)$ as equivalent to the asymptotic invariance of $θ$ under $μ$ in the weak* topology. We apply these results to provide a probabilistic proof of the Choquet-Deny theorem for nilpotent groups, among other applications.

On the inclusion of bounded harmonic functions of random walks

TL;DR

The paper addresses when spaces of bounded harmonic functions for two random-walk measures on a group satisfy . It develops a martingale-based approach built on asymptotic commutativity: if and , then the inclusion holds, with equality when and commute and generate . The authors extend the framework to general Markov chains, provide a Derriennic-type necessary-and-sufficient characterization via total variation and weak*-convergence on boundary spaces, and relate these conditions to hitting models of the Poisson boundary. They then derive several applications, including a probabilistic proof of the Choquet-Deny theorem for nilpotent groups and structural results about the center and product groups. Overall, the work bridges probabilistic martingale methods with boundary theory to obtain robust inclusions and equivalences for harmonic functions beyond locally compact second countable groups.

Abstract

We investigate the conditions under which the space of bounded harmonic functions of a probability measure on a group is contained in that of another measure . We establish that asymptotic commutativity, defined by the condition as , is sufficient to guarantee the inclusion , provided is absolutely continuous with respect to a convex combination of convolution powers of . By employing martingale convergence techniques rather than ergodic-theoretic arguments, we demonstrate that this result holds without topological assumptions on (such as local compactness) and extends to general Markov chains. Furthermore, utilizing hitting models for the Poisson boundary, we characterise the inclusion as equivalent to the asymptotic invariance of under in the weak* topology. We apply these results to provide a probabilistic proof of the Choquet-Deny theorem for nilpotent groups, among other applications.
Paper Structure (14 sections, 20 theorems, 109 equations)

This paper contains 14 sections, 20 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Asymptotically commuting measures
  4. Proof of Theorem \ref{['thm:mainA']}
  5. General Markov chains
  6. Necessary and sufficient condition
  7. $\mu$-Boundaries as hitting measures
  8. The weak* condition
  9. The total-variation condition
  10. Concluding the sufficient and necessary condition
  11. Applications
  12. The centre and bounded harmonic functions
  13. Convex combinations of convolution powers
  14. Restricting bounded harmonic functions on $G_1\times G_2$

Key Result

Theorem A

(Theorem thm:asmpt-commuting) Let $G$ be a group equipped with a $G$-invariant $\sigma$-algebra $\Sigma$ and $\mu$, $\theta$ be two probability measures on $(G,\Sigma)$ such that $\theta\ll\sum_{n\geq0}\frac{1}{2^{n+1}}\mu^{*n}$ and where $*$ stands for convolution and norm is the total-variation norm. Then ${H}^\infty (G,\mu)\subseteq {H}^\infty (G,\theta)$.

Theorems & Definitions (43)

  • Theorem A
  • Theorem B
  • Theorem 2.1: Furstenberg, Furst71
  • Definition 2.2
  • Definition 3.1
  • Example 3.2
  • Proposition 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 33 more