Convexity at infinity and Brownian motion on manifolds with unbounded negative curvature
In the paper harmonic functions on simply connected complete Riemannian manifolds with curvature unbounded from below are under consideration. The problem is treated in probabilistic terms by the description of the behaviour of the Brownian motion at infinity. Note that the above manifold with curvature bounded from below is stochastically complete (this was shown by K. D. Elworthy on the basis of some result by Yau on harmonic functions). The main result of the paper is an example of a 3- dimensional simply connected complete Riemannian manifold with negative curvature \((\leq -1)\) where the behaviour of the Brownian motion at infinity is absolutely different from that of the bounded case. In spite of the fact that the Brownian motion has a.s. infinite lifetime (i.e. the manifold is stochastically complete), it exits from the manifold a.s. at a single point of the infinity sphere. A variant of the method gives also that any point of the infinity sphere is a.s. a cluster point of the Brownian motion. These results are also extended to higher dimensions.
scientific article; zbMATH DE number 572268