Exponentials and $R$-recurrent random walks on groups

TL;DR

This paper investigates the relationship between $R$-recurrent random walks, exponential functions, and group recurrence properties on locally compact groups, establishing conditions for their existence and uniqueness.

Contribution

It introduces a connection between $r$-invariant measures, continuous exponentials, and $R$-recurrence for random walks on groups, extending previous theoretical frameworks.

Findings

Existence of a unique continuous exponential associated with the random walk.

Characterization of $R$-recurrence in terms of group recurrence and Harris random walks.

Conditions under which $r$-invariant measures lead to $R$-recurrent random walks.

Abstract

On a locally compact group $E$ with countable base, we consider a random walk $X$ that has a unique (up to a positive factor) $r$-invariant measure for some $r>0$. Under some weak conditions on the measure, there is a unique continuous exponential on $E$ naturally associated with $X$. It follows that there exists an $R$-recurrent random walk in the sense of Tweedie on $E$ if and only if $E$ is a recurrent group and there exists a Harris random walk on~$E$.