SOME ASYMPTOTIC PROPERTIES OF RANDOM WALKS ON HOMOGENEOUS SPACES

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 ? subset of G, the mu-walk and the geodesic flow on ?\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.