Random Walks with Bounded First Moment on Finite-volume Spaces - Bénard, Timothée; Saxcé, Nicolas de (Geometric and functional analysis, 2022)
Let G be a real Lie group, Λ ≤ G a lattice, and Ω = G / Λ . We study the equidistribution properties of the left random walk on Ω induced by a probability measure μ on G . It is assumed that μ has a finite first moment, and that the Zariski closure of the group generated by the support of μ in the adjoint representation is semisimple without compact factors. We show that for every starting point x ∈ Ω , the 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.