Harmonic functions on homogeneous spaces

Given a locally compact group G acting on a locally compact space X and a probability measure sigma on G, a real Borel function f on X is called sigma-harmonic if it satisfies the convolution equation f = sigma*f. We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of sigma-admissible neighbourhoods of the identity, relative to X, then every bounded sigma-harmonic function on X is constant. Consequently, for spread out sigma, the bounded sigma-harmonic functions are constant on each connected component of a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded sigma-harmonic functions on X are constant which extends Furstenberg's result for connected semisimple Lie groups.