  In this paper we extend the construction of random walks with a prescribed
Poisson boundary to the case of measures in the class of a generalized Gibbs
state. The price for dropping the $\alpha$-quasiconformal assumptions is that
we must restrict our attention to CAT($-\kappa$) groups. Apart from the new
estimates required, we prove a new approximation scheme to provide a positive
basis for positive functions in a metric measure space.
