Convergence of martingales on manifolds of negative curvature
type: Item
instance of: scholarly article
title: Convergence of martingales on manifolds of negative curvature
zbMATH Open document ID: 0565.60042
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1985-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77254
review text: Let \(X_ t\) be a martingale on a Riemannian manifold M with sectional curvatures bounded from above by \(-k^ 2<0.\) If \(X_ t\) moves in at least two 'random directions' with not too different never vanishing intensities and \(X_ t\) does not escape from unit balls too fast, then \(X_{\infty}=\lim_{t\to \infty}X_ t\) exists a.s. in the sphere at infinity (the set of directions of geodesic rays) (Theorem A). When \(X_ t\) is Brownian motion Theorem A gives a.s. convergence to a limit in the sphere at infinity if the sectional curvatures of M do not decay to minus infinity too fast with radial distance (Theorem B). This implies the existence of non-constant bounded harmonic functions on M, namely \(f(x)=E_ x(h(X_{\infty}))\) with \(h\in C(S^{d-1})\). For the proof of Theorem B a lower bound for the probability that Brownian motion remains in a ball of radius a up to time t is established for the sectional curvatures of M being pinched between two negative constants.
reviewed by: Hans Crauel
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 31C12
zbMATH DE Number: 3901750
zbMATH Keywords: martingale on a Riemannian manifold
zbMATH Keywords: sectional curvatures
zbMATH Keywords: existence of non-constant bounded harmonic functions
author: Richard W. R. Darling
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3901750