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 $\Lambda \subseteq G$, 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.