Harmonicity of quasiconformal measures and Poisson boundaries of hyperbolic spaces

Abstract

We consider a group G of isometries acting on a (not necessarily geodesic) delta-hyperbolic space X and possessing a radial limit set of full measure within its limit set. For any continuous quasiconformal measure w supported on the limit set, we produce a stationary measure m on G. Moreover the limit set together with w forms a m-boundary and w is harmonic with respect to the random walk induced by m. In the case when X is a CAT(-1) space and G acts cocompactly, for instance, we show that m has finite first moment. This implies that the boundary of X with w is the unique Poisson boundary for m. As a bi-product, we establish sufficient conditions for a set of continuous functions to form a positive basis, either in the L^1 or sup norm, for the space of uniformly positive lower-semicontinuous functions on a general metric measure space.