Regularity of the drift for random walks in groups acting on Gromov hyperbolic spaces

TL;DR

This paper establishes the continuity and large deviation principles for the drift of random walks on groups acting on Gromov hyperbolic spaces, extending existing ergodic theorems to non-proper spaces.

Contribution

It refines the multiplicative ergodic theorem for Gromov hyperbolic spaces and proves new results on drift regularity and large deviations for non-proper spaces.

Findings

Proved continuity of the drift for random walks in Gromov hyperbolic spaces.

Established large deviation principles for the drift.

Extended ergodic theorems to non-proper hyperbolic spaces.

Abstract

In this work we prove the continuity and existence of large deviations for the drift of random walks on groups acting by isometries on Gromov Hyperbolic Spaces. Through the process we refine the multiplicative ergodic theorem of Karlsson and Gou\"ezel for such spaces. The works goes beyond what is known in the literature by allowing spaces that are not necessarily proper.