# Besov spaces and random walks on a hyperbolic group: boundary traces and reflecting extensions of Dirichlet forms

## TL;DR

This paper explores the boundary behavior of random walks on hyperbolic groups with conformal dimension less than 2, introducing Besov spaces and analyzing harmonic measures through Dirichlet forms.

## Contribution

It establishes a connection between boundary traces of random walks, Besov spaces, and cohomology, and introduces a reflecting extension of Dirichlet forms for hyperbolic groups.

## Key findings

- Existence of a boundary trace process at infinity for certain hyperbolic groups
- Isomorphism between newly introduced Besov spaces and cohomology-based Besov spaces
- Analysis of harmonic measure regularity using Dirichlet form potential theory

## Abstract

We show the existence of a trace process at infinity for random walks on hyperbolic groups of conformal dimension < 2 and relate it to the existence of a reflecting random walk. To do so, we employ the theory of Dirichlet forms which connects the theory of symmetric Markov processes to functional analytic perspectives. We introduce a family of Besov spaces associated to random walks and prove that they are isomorphic to some of the Besov spaces constructed from the co-homology of the group studied in Bourdon-Pajot (2003). We also study the regularity of harmonic measures of random walks on hyperbolic groups using the potential theory associated to Dirichlet forms.