Random Walks with Bounded First Moment on Finite-volume Spaces

Abstract

AbstractLet G be a real Lie group, $$\Lambda \le G$$ Λ ≤ G a lattice, and $$\Omega =G/\Lambda $$ Ω = G / Λ . We study the equidistribution properties of the left random walk on $$\Omega $$ Ω induced by a probability measure $$\mu $$ μ on G. It is assumed that $$\mu $$ μ has a finite first moment, and that the Zariski closure of the group generated by the support of $$\mu $$ μ in the adjoint representation is semisimple without compact factors. We show that for every starting point $$x\in \Omega $$ x ∈ Ω , the $$\mu $$ μ -walk with origin x has no escape of mass, and equidistributes in Cesàro averages toward some homogeneous measure. This extends several fundamental results due to Benoist-Quint and Eskin-Margulis for walks with finite exponential moment.