The radial part of a \(\Gamma\)-martingale and a non-implosion theorem
type: Item
instance of: scholarly article
title: The radial part of a \(\Gamma\)-martingale and a non-implosion theorem
zbMATH Open document ID: 0839.58062
author: Wilfrid S. Kendall
published in: The Annals of Probability
publication date: 1996-06-24T00:00:00Z
review text: A stochastic process \(X\) taking values in a complete Riemannian manifold is a \(\Gamma\)-martingale if its compositions with \(C^2\) maps are submartingales at least when \(X\) belongs to the region of convexity of the map. For time approaching zero such a process has always a limit which is finite or infinite. The aim of the paper is to give conditions for this limit to be finite, the property which implies that the corresponding diffusion is Feller. To this end behaviour of the radial part \(R\) of the process \(X\) is analyzed. It is shown that if the manifold has sectional curvatures bounded from below by \(- K^2\) then \(R\) is a semimartingale with the semimartingale decomposition \(dR = dN + d \Lambda\), where the local martingale \(N\) satisfies \(d[N,N] \leq \text{trace} d [X,X]\) and the finite variation part has the property \[ I_{(R > 0)} d \Lambda \leq {K \over 2}\quad \text{coth} (KR)\quad \text{trace} d [X,X]. \] This result allows to extend the approach of Hsu to obtain the integral curvature condition which insures that \(X(0 + ) \neq \infty\).
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 60H99
zbMATH DE Number: 796507
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: comparison theorems
zbMATH Keywords: Riemannian Brownian motion
zbMATH Keywords: implosion
zbMATH Keywords: convexity
zbMATH Keywords: Feller property
reviewed by: B. Goldys
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176988276
OpenAlex ID: W2038172114
DOI: 10.1214/AOP/1176988276
description: scientific article; zbMATH DE number 796507