Some asymptotic properties of random walks on homogeneous spaces

TL;DR

This paper studies the asymptotic behavior of random walks on homogeneous spaces of semisimple Lie groups, showing they tend to stay near certain Weyl chambers and relating their recurrence properties to geodesic flows.

Contribution

It extends classical recurrence and ergodicity results from Brownian motion to $ ext{random walks}$ on homogeneous spaces of rank one semisimple Lie groups.

Findings

Random trajectories stay near specific Weyl chambers.

Recurrence and ergodicity of the walk match those of geodesic flow in rank one.

Extension of Hopf-Tsuji-Sullivan-Kaimanovich theorem to random walks.

Abstract

Let $G$ be a connected semisimple real Lie group with finite center, and $\mu$ a probability measure on $G$ whose support generates a Zariski-dense subgroup of $G$. We consider the right $\mu$-random walk on $G$ and show that each random trajectory spends most of its time at bounded distance of a well-chosen Weyl chamber. We infer that if $G$ has rank one, and $\mu$ has a finite first moment, then for any discrete subgroup $\Lambda \subseteq G$, the $\mu$-walk and the geodesic flow on $\Lambda \backslash G$ are either both transient, or both recurrent and ergodic, thus extending a well known theorem due to Hopf-Tsuji-Sullivan-Kaimanovich dealing with the Brownian motion.