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

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 reflected 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.