Random walks with bounded first moment on finite-volume spaces

TL;DR

This paper investigates the equidistribution of random walks with bounded first moment on finite-volume spaces derived from Lie groups, extending key results to broader moment conditions and demonstrating convergence to homogeneous measures.

Contribution

It extends fundamental equidistribution results for random walks on homogeneous spaces to cases with only finite first moment, without requiring exponential moment conditions.

Findings

No escape of mass for the random walk starting from any point.

Cesàro averages of the walk converge to homogeneous measures.

Results generalize previous theorems to weaker moment assumptions.

Abstract

Let $G$ be a real Lie group, $\Lambda\leq G$ 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 $x\in \Omega$, the $\mu$-walk with origin $x$ has no escape of mass, and equidistributes in Ces\`aro averages toward some homogeneous measure. This extends several fundamental results due to Benoist-Quint and Eskin-Margulis for walks with finite exponential moment.