The limiting angle of certain riemannian brownian motions

Abstract

Abstract 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 Sect M (x) denote the sectional curvature of any plane section in M x . We prove that for each c > 2, there is a 0 < β < 1 such that if ‐ L 2 r(x) 2β ≦ Sect M (x) ≦ ‐cr(x) −2 for all x in the complement of a compact set, then lim t → ∞ θ( X t ) 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.