Random Walks with Bounded First Moment on Finite-volume Spaces

• Published in: Geometric and functional analysis Vol. 32; no. 4; pp. 687 - 724
• Main Authors: Bénard, Timothée, Saxcé, Nicolas de
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.08.2022; Springer Nature B.V
• Subjects: Analysis; Lie groups; Mathematics; Mathematics and Statistics; Random walk; 37A50; Recurrence; Homogeneous spaces; Secondary 60G07; Random walks; Primary 37B20; 22F30
• ISSN: 1016-443X; 1420-8970
• DOI: 10.1007/s00039-022-00607-6

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.