Brownian motion and Dirichlet problems at infinity

Abstract

We discuss angular convergence of Riemannian Brownian motion on a Cartan--Hadamard manifold and show that the Dirichlet problem at infinity for such a manifold is uniquely solvable under the curvature conditions $-Ce^{(2-\eta) ar(x)}\le K_M(x)\le-a^2$\vspace*{0.5pt} ($\eta>0$) and $-Cr(x)^{2\beta} \le K_M(x)\le - \alpha (\alpha-1)/r(x)^2$ ($\alpha>\beta+2>2$), respectively.