Random walks of infinite moment on free semigroups

TL;DR

This paper investigates random walks on free semigroups, introducing the w-logarithmic moment concept and characterizing their Poisson boundaries even when classical entropy conditions are not met.

Contribution

It introduces the w-logarithmic moment and extends Poisson boundary identification to measures lacking finite entropy.

Findings

Poisson boundary identified for measures with finite entropy or w-logarithmic moment

w-logarithmic moment generalizes classical entropy conditions

Characterization of random walks with infinite moment on free semigroups

Abstract

We consider random walks on finitely or countably generated free semigroups, and identify their Poisson boundaries for classes of measures which fail to meet the classical entropy criteria. In particular, we introduce the notion of w-logarithmic moment, and we show that if a random walk on a free semigroup has either finite entropy or finite w-logarithmic moment for some word w, then the space of infinite words with the resulting hitting measure is the Poisson boundary.