Random Walks with Bounded First Moment on Finite-volume Spaces

Abstract

Let G be a real Lie group, Lambda le G a lattice, and Omega =G/Lambda . We study the equidistribution properties of the left random walk on Omega induced by a probability measure mu on G. It is assumed that mu has a finite first moment, and that the Zariski closure of the group generated by the support of mu in the adjoint representation is semisimple without compact factors. We show that for every starting point xin Omega , the mu -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.