Some asymptotic properties of random walks on homogeneous spaces

DOI10.3934/JMD.2023004arXiv2104.13181MaRDI QIDQ6366244

Timothée Bénard

Publication date: 27 April 2021

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 $LambdasubseteqG$, the $mu$-walk and the geodesic flow on $$ 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.

Mathematics Subject Classification ID

Sums of independent random variables; random walks (60G50) Differential geometry of homogeneous manifolds (53C30) Ergodic theory on groups (22D40) Discrete subgroups of Lie groups (22E40) Dynamical systems and their relations with probability theory and stochastic processes (37A50) Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents (37H15) Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) (37D40) Homogeneous flows (37A17)

Related Items (1)

Expanding measures: Random walks and rigidity on homogeneous spaces

This page was built for publication: Some asymptotic properties of random walks on homogeneous spaces

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6366244)