Some asymptotic properties of random walks on homogeneous spaces

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.