Some asymptotic properties of random walks on homogeneous spaces. (English) Zbl 1530.37008
Summary: 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 [E. Hopf, Bull. Am. Math. Soc. 77, 863–877 (1971; Zbl 0227.53003); M. Tsuji, Potential theory in modern function theory. Tokyo: Maruzen Co (1959; Zbl 0087.28401); D. Sullivan, Ann. Math. Stud. None, 465–496 (1981; Zbl 0567.58015); V. A. Kaimanovich, Ann. Math. (2) 152, No. 3, 659–692 (2000; Zbl 0984.60088)] dealing with the Brownian motion.
MSC:
| 37A17 | Homogeneous flows |
| 37A50 | Dynamical systems and their relations with probability theory and stochastic processes |
| 37H15 | Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents |
| 37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |
| 53C30 | Differential geometry of homogeneous manifolds |
| 22D40 | Ergodic theory on groups |
| 22E40 | Discrete subgroups of Lie groups |
| 60G50 | Sums of independent random variables; random walks |
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