Equidistribution of mass for random processes on finite-volume spaces

Abstract

Let G be a real Lie group, Λ ⊆ G a lattice, and X = G/Λ. We fix a probability measure μ on G and consider the left random walk induced on X. It is assumed that μ is aperiodic, has a finite first moment, spans a semisimple algebraic group without compact factors, and has two non mutually singular convolution powers. We show that for every starting point x ∈ X, the n-th step distribution μn*δx of the walk weak-* converges toward some homogeneous probability measure on X.