Recurrence and ergodicity of random walks on linear groups and on homogeneous spaces

TL;DR

This paper investigates the recurrence and ergodicity of random walks on linear groups and homogeneous spaces, establishing conditions related to group growth and analyzing the structure of subgroups and actions.

Contribution

It provides a characterization of recurrence for random walks on subgroups of linear groups over local fields based on quadratic growth, and analyzes ergodicity properties on homogeneous spaces.

Findings

Recurrent random walks occur only on groups with at most quadratic growth.

Detailed analysis of ergodicity for specific random walks on homogeneous spaces.

Structural insights into subgroups of linear groups over local fields.

Abstract

We discuss recurrence and ergodicity properties of random walks and associated skew products for large classes of locally compact groups and homogeneous spaces. In particular we show that a closed subgroup of a product of finitely many linear groups over local fields supports a recurrent random walk if and only if it has at most quadratic growth. We give also a detailed analysis of ergodicity properties for special classes of random walks on homogeneous spaces. The structure of closed subgroups of linear groups over local fields and the properties of group actions with respect to stationary measures play an important role in the proofs.