Spread Out Random Walks on Homogeneous Spaces

TL;DR

This paper analyzes spread out random walks on homogeneous spaces, revealing their asymptotic behavior, equidistribution, and classical limit theorems, with new results on recurrence and ratio limits in infinite volume cases.

Contribution

It provides a comprehensive analysis of spread out random walks on homogeneous spaces, including asymptotics, equidistribution, and recurrence, settling a long-standing conjecture.

Findings

Finite volume spaces show exponential equidistribution towards Haar measure.

Classical limit theorems hold for spread out random walks.

Recurrence and ratio limit theorems are established for infinite volume cases.

Abstract

A measure on a locally compact group is called spread out if one of its convolution powers is not singular with respect to Haar measure. Using Markov chain theory, we conduct a detailed analysis of random walks on homogeneous spaces with spread out increment distribution. For finite volume spaces, we arrive at a complete picture of the asymptotics of the $n$-step distributions: They equidistribute towards Haar measure, often exponentially fast and locally uniformly in the starting position. In addition, many classical limit theorems are shown to hold. In the infinite volume case, we prove recurrence and a ratio limit theorem for symmetric spread out random walks on homogeneous spaces of at most quadratic growth. This settles one direction in a long-standing conjecture.