Convexity at infinity and Brownian motion on manifolds with unbounded negative curvature
type: Item
instance of: scholarly article
title: Convexity at infinity and Brownian motion on manifolds with unbounded negative curvature
zbMATH Open document ID: 0804.58056
DOI: 10.4171/RMI/150
author: Alano Ancona
published in: Revista Matemática Iberoamericana
publication date: 1995-01-12T00:00:00Z
full work available at URL: https://eudml.org/doc/39456
review text: 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.
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 58E20
zbMATH DE Number: 572268
zbMATH Keywords: Riemannian manifolds
zbMATH Keywords: behaviour at infinity
zbMATH Keywords: Brownian motion
reviewed by: Yuri E. Gliklikh
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 572268