The Unique stationary boundary property

TL;DR

This paper establishes the equivalence between the uniqueness of the stationary measure on the universal Poisson boundary and the isomorphism of this boundary with the universal minimal strongly proximal G-flow for second countable locally compact groups.

Contribution

It proves that for such groups, the uniqueness of the stationary measure characterizes the boundary as the universal minimal strongly proximal G-flow.

Findings

The measure $ u$ is unique if and only if the boundary is isomorphic to the universal minimal strongly proximal G-flow.

Provides a characterization of stationary measures in terms of boundary isomorphism.

Extends understanding of boundary properties in the context of locally compact groups.

Abstract

Let $G$ be a locally compact group and $\mu$ an admissible probability measure on $G$. Let $(B,\nu)$ be the universal topological Poisson $\mu$-boundary of $(G,\mu)$ and $\Pi_s(G)$ the universal minimal strongly proximal $G$-flow. This note is inspired by a recent result of Hartman and Kalantar. We show that for a locally compact second countable group $G$ the following conditions are equivalent: (i) the measure $\nu$ is the unique $\mu$-stationary measure on $B$, (ii) $(B,G) \cong (\Pi_s(G),G)$.