Conditioned superprocesses and their weighted occupation times
type: Item
instance of: scholarly article
title: Conditioned superprocesses and their weighted occupation times
zbMATH Open document ID: 0815.60078
DOI: 10.1016/0167-7152(94)00050-I
author: Stephen M. Krone
published in: Statistics \& Probability Letters
publication date: 1995-02-22T00:00:00Z
review text: Consider a superprocess \(X\) (an \(M_ F(E)\)-valued Markov process with Laplace functional, where \(M_ F(E)\) denotes the space of finite measures on a locally compact separable metric space \(E\) with its Borel \(\sigma\)-field). Because of the fact that the initial measure \(\mu\) of \(X_ t\) is finite, the process \(X_ t\) will die in finite time, \({\mathbf P}^ \mu\)-a.s., in other words, \(X_ t = 0\) for sufficiently large \(t\). So-called conditioned superprocesses, obtained by starting \(X\) at an initial measure \(\mu \in M_ F(E) \setminus \{0\}\) and conditioning on the event that the process stays alive forever, are considered. Then the asymptotic behavior of the weighted occupation time (whether the total amount of mass which ever visits a fixed set is finite or infinite) is studied. Necessary and sufficient conditions, based on the asymptotics of the underlying motion, for the total occupation time to be infinite are given. Some special cases are investigated. It is also proved that, when properly scaled, the occupation times for conditioned super-Brownian motion on \({\mathbf R}^ d\) converge to a function of a local time when \(d \leq 3\).
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60J55
zbMATH DE Number: 724095
zbMATH Keywords: measure-valued processes
zbMATH Keywords: superprocesses
zbMATH Keywords: weighted occupation times
zbMATH Keywords: local times
MaRDI profile type: Publication
cites work: Q3808983
cites work: Q3713281
cites work: Q3721531
cites work: Two representations of a conditioned superprocess
cites work: Measure-valued Markov branching processes conditioned on non-extinction
cites work: Construction and regularity of measure-valued Markov branching processes
cites work: A weighted occupation time for a class of measure-valued branching processes
cites work: Ergodic theory and a local occupation time for measure-valued critical branching brownian motion
cites work: Comparison methods for a class of function valued stochastic partial differential equations
cites work: Local times for superdiffusions
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Q3997782
cites work: Q4734585
cites work: Some properties for the measure-valued branching diffusion processes
full work available at URL: 0167-7152(94)00050-i
OpenAlex ID: W1995211764
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description: scientific article; zbMATH DE number 724095