Random walks and rank one isometries on CAT(0) spaces

TL;DR

This paper studies random walks on groups acting on CAT(0) spaces, proving almost sure convergence to rank one boundary points, uniqueness of stationary measures, and positive drift under non-elementary actions with rank one elements.

Contribution

It establishes convergence, uniqueness, and positivity results for random walks on CAT(0) spaces with rank one isometries, extending understanding of boundary behavior.

Findings

Random walks converge almost surely to rank one boundary points.

There is a unique stationary measure on the boundary.

The random walk has almost surely positive drift.

Abstract

Let $G$ be a discrete group, $\mu$ a measure on $G$ and $X$ a proper CAT(0) space. We show that if $G$ acts non-elementarily with a rank one element on $X$, then the pushforward $\{Z_n o \}_n$ to $X$ of the random walk generated by $\mu$ converges almost surely to a rank one point of the boundary. We also show that in this context, there is a unique stationary measure on the visual boundary $\partial_\infty X$ of $X$, and that the drift of the random walk is almost surely positive.