Transience in law for symmetric random walks in infinite measure

TL;DR

This paper investigates the behavior of symmetric random walks in infinite measure spaces, demonstrating conditions under which escape of mass occurs and providing a converse to a known recurrence theorem.

Contribution

It establishes a new criterion for escape of mass in symmetric random walks and extends the Eskin-Margulis recurrence theorem to infinite volume spaces.

Findings

Escape of mass occurs for almost every starting point under certain symmetry conditions.

Invariant sets have zero or infinite measure, influencing walk behavior.

Provides a converse to Eskin-Margulis recurrence theorem in infinite measure spaces.

Abstract

We consider a random walk on a second countable locally compact topological space endowed with an invariant Radon measure. We show that if the walk is symmetric and if every subset which is invariant by the walk has zero or infinite measure, then one has escape of mass for almost every starting point. We then apply this result in the context of homogeneous random walks on infinite volume spaces, and deduce a converse to Eskin-Margulis recurrence theorem.