  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.
