Chris Connell (Indiana University)

Prescribing boundaries for random walks on hyperbolic groups

Abstract. For a random walk on a group, Furstenberg associated to it a Poisson boundary; a measure theoretic representation space for bounded harmonic functions. For a Gromov hyperbolic group $G$ with geodesic boundary $B$ , we consider the inverse problem of starting with a measure $m$ on $B$ and asking whether or not $(B,m)$ can arise as a Poisson boundary for some random walk on $G$ . For $G$ a CAT(-1) group and m any Lipschitz quasiconformal measure, or more generally a measure from a large family of classes of conditional Gibbs states, we give an affirmative answer. For general groups acting on Gromov hyperbolic spaces we show that continuous quasiconformal measures on the ``radial" limit set arise as $G$ -quotients of the Poisson boundary. The proofs also establish approximation results generalizing those of Bonsall and Hayman-Lyons. This is joint work with Roman Muchnik.