Noncommutative Ergodic Theorems

TL;DR

This paper explores the asymptotic behavior of ergodic products of isometries in metric spaces, revealing conditions for diffusion or preferred boundary directions, with applications to various stochastic processes.

Contribution

It provides new results on noncommutative ergodic theorems, extending classical results to nonintegrable functions, group random walks, and Brownian motion on manifolds.

Findings

Either sublinear diffusion occurs or a preferred boundary direction exists

Results apply to nonintegrable functions and stochastic processes

Comparison with classical ergodic theorems enhances understanding of asymptotic behavior

Abstract

We present recent results about the asymptotic behavior of ergodic products of isometries of a metric space X. If we assume that the displacement is integrable, then either there is a sublinear diffusion or there is, for almost every trajectory in X, a preferred direction at the boundary. We discuss the precise statement when X is a proper metric space and compare it with classical ergodic theorems. Applications are given to ergodic theorems for nonintegrable functions, random walks on groups and Brownian motion on covering manifolds.