Harmonicity of Gibbs measures

Abstract

We show that any continuous measure $\nu$ in the class of a generalized Gibbs stream on the boundary of a CAT($-\kappa$) group $G$ arises as a harmonic measure for a random walk on $G$. Under an additional mild hypothesis on $G$ and for $\nu$, Hölder equivalent to a Gibbs measure, we show that $(\partial G,\nu )$ arises as a Poisson boundary for a random walk on $G$. We also prove a new approximation theorem for general metric measure spaces giving quite flexible conditions for a set of functions to be a positive basis for the cone of positive continuous functions