2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/75653In 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 $α$-quasiconformal assumptions is that we must restrict our attention to CAT($-κ$) 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.Group TheoryDynamical SystemsProbability60J50; 20F67; 37A35; 41A65text