On uniqueness of invariant measures for random walks on HOMEO(R)

TL;DR

This paper investigates the conditions under which a random walk on the group of orientation-preserving homeomorphisms of the real line has a unique invariant Radon measure, extending previous results to broader classes of systems.

Contribution

It establishes the uniqueness of invariant measures for a wider class of random walks on homeomorphisms of the real line under recurrence, contraction, and unbounded action conditions.

Findings

Proves uniqueness of invariant Radon measure under specified conditions

Extends classical results to broader classes of homeomorphism systems

Provides a framework for analyzing invariant measures in non-strongly contractive systems

Abstract

We consider random walks on the group of orientation-preserving homeomorphisms of the real line ${\mathbb R}$. In particular, the fundamental question of uniqueness of an invariant measure of the generated process is raised. This problem was already studied by Choquet and Deny (1960) in the context of random walks generated by translations of the line. Nowadays the answer is quite well understood in general settings of strongly contractive systems. Here we focus on broader class of systems satisfying the conditions: recurrence, contraction and unbounded action. We prove that under these conditions the random process possesses a unique invariant Radon measure on ${\mathbb R}$. Our work can be viewed as a subsequent paper of Babillot et al. (1997) and Deroin et al. (2013).