The behavior of superprocesses near extinction
type: Item
instance of: scholarly article
title: The behavior of superprocesses near extinction
zbMATH Open document ID: 0749.60046
published in: The Annals of Probability
publication date: 1992-06-28T00:00:00Z
review text: Let \(\{X_ t;\;t\geq 0\}\) be a continuous superprocess in a locally compact separable metric space \(E\) starting with a finite measure. The underlying motion process is assumed to be a conservative Feller process. A martingale problem characterization is used to study the behavior of \(X\) near its extinction time. Roughly speaking, \(X_ t\) normalized to be a probability measure will extinct at a single point distributed according to the motion law. In some cases, there are times near extinction at which the closed support of the superprocess is concentrated near the extinction point (although the closed support of \(X_ t\) at fixed times \(t\) is typically spread throughout space). In the case of a super-\(\alpha\)-stable motion \(X\) in \(E=\mathbb{R}\), \(0<\alpha\leq 2\), a Tanaka-like formula for \((X_ t,f)\) is obtained and discussed for a class of functions \(f\) including the indicator functions \(\mathbf{1}\{x:\;x\leq a\}\), \(a\in\mathbb{R}\), of half axes. For instance, \(\{X_ t(x\leq a);\;t\geq 0\}\) fails to be a semimartingale if \(1<\alpha\leq 2\).
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60G44
zbMATH DE Number: 34055
zbMATH Keywords: measure-valued process
zbMATH Keywords: Feller process
zbMATH Keywords: superprocess
zbMATH Keywords: Tanaka-like formula
zbMATH Keywords: semimartingale
author: Roger Tribe
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176989927
OpenAlex ID: W2024346635
DOI: 10.1214/AOP/1176989927
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description: scientific article; zbMATH DE number 34055