Limiting angle of Brownian motion on certain manifolds (Q1924279)

Let \(M\) be a simply connected complete \(m\)-dimensional Riemannian manifold with nonpositive sectional curvature (then \(M\) is diffeomorphic to \(\mathbb{R}^m)\). Denote by \(r(x)\) the Riemannian distance between 0 and \(x\). The author supposes that outside a compact set the sectional curvatures are bounded by \(-cr^{-2} (x)\) from above and by \(-\widetilde cr^2(x)\) from below where \(c\) and \(\widetilde c\) are positive constants \((c> 3/4\) for \(m= 3)\) and proves that under these assumptions the angular part of a Brownian motion on \(M\) tends to a limit as time tends to infinity, and the closure of support of the distribution of this limit is the entire \(S^{m-1}\).