Equidistribution of mass for random processes on finite-volume spaces

TL;DR

This paper proves that for certain random walks on spaces formed by Lie groups and lattices, the distribution of the walk's position becomes uniformly spread out over time, approaching a homogeneous measure.

Contribution

It establishes the equidistribution of mass for random processes on finite-volume spaces under broad conditions, extending previous results to more general settings.

Findings

Weak-* convergence of distributions to homogeneous measures

Conditions under which random walks become equidistributed

Applicability to semisimple algebraic groups without compact factors

Abstract

Let $G$ be a real Lie group, $\Lambda\subseteq G$ a lattice, and $X=G/\Lambda$. We fix a probability measure $\mu$ on $G$ and consider the left random walk induced on $X$. It is assumed that $\mu$ 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 $x\in X$, the $n$-th step distribution $\mu^n*\delta_{x}$ of the walk weak-$\ast$ converges toward some homogeneous probability measure on $X$.