Limiting angle of certain Riemannian Brownian motions

Let M be a Cartan-Hadamard manifold of dimension d ≧ 3, let p ϵ M and x = exp {r(x)θ(x)} be geodesic polar coordinates with pole p and let X be the Brownian motion on M. Let SectM(x) denote the sectional curvature of any plane section in Mx. We prove that for each c > 2, there is a 0 <β <1 such that if - L2r(x)2β ≦ SectM(x) ≦ -cr(x)−2 for all x in the complement of a compact set, then limt → ∞ θ(Xt) exists a.s. and defines a nontrivial invariant random variable. The Dirichlet problem at infinity and a conjecture of Greene and Wu are also discussed.