Jump processes on the boundary of a hyperbolic group

On a non elementary, Gromov hyperbolic group of conformal dimension less than two - say a surface group - we consider a symmetric probability measure whose support generates the whole group and with a finite second moment. To such a probability mesaure, we associate a Besov space on the boundary of the group. All the Besov spaces constructed that way turn out to be isomorphic. They are also isomorphic to the Besov spaces already constructed by M. Bourdon and H. Pajot using the cohomology of the group. In our work, cohomology spaces are replaced by martingales.

We also provide a probabilistic interpretation of the Besov spaces as the Dirichlet spaces of the trace on the boundary of the random walks reflected on the boundary. Along with the definition of Besov spaces, come notions of sets of zero capacity, smooth measures and measures of finite energy on the boundary. Using heat kernel estimates, we obtain an integral criterion for a measure on the boundary to have finite energy.

We apply this criterion and deviation inequalities to show that harmonic measures of random walks with a driving measure with a finite first moment are smooth measures in our sense, thus obtaining a quite general regularity property of harmonic measures.

P. Mathieu (Aix-Marseille) and Tokushige Y. (Kyoto)