THE POISSON BOUNDARY FOR RANK-ONE MANIFOLDS AND THEIR COCOMPACT LATTICES

Let M be a complete simply connected Riemannian manifold of non-positive curvature. If M admits a compact quotient and is of rank one, then Brownian motion converges at partial derivative M, the sphere at infinity, and defines the family of harmonic measures on partial derivative M. In this paper it is shown that partial derivative M together with this family of measures is naturally isomorphic to the Poisson boundary of M.