Local extinction versus local exponential growth for spatial branching processes.
type: Item
instance of: scholarly article
title: Local extinction versus local exponential growth for spatial branching processes.
zbMATH Open document ID: 1056.60083
author: János Engländer
author: A. E. Kyprianou
published in: The Annals of Probability
publication date: 2004-09-15T00:00:00Z
review text: Let \(X\) be either the branching diffusion with generator \(Lu+ \beta(u^2-u)\) on a domain \(D\) in \(R^d\), \(\beta\) nonnegative, bounded, and not identically \(0\), or the superprocess with generator \(Lu+\beta u-\alpha u^2\), \(\alpha\) positive and \(\beta\) bounded from above. Let the starting measure \(\mu\) be not identically vanishing, finite, with support compactly embeddable in \(D\). \(X\) is said to exhibit weak local extinction under \(P_\mu\), if \(P_\mu(\lim\sup_{t\uparrow \infty} X_t(B)=0)=1\) for every Borel set \(B\) embeddable in \(D\). The process \(X\) exhibits weak local extinction if and only if the generalized principal eigenvalue \(\lambda_c\) of \(L+\beta\) is nonpositive. In particular, this property does not depend on the starting measure. If \(\lambda_c>0\), then for any \(\lambda<\lambda_c\) and any nonempty set \(B\) compactly embeddable in \(D\), \[ P_\mu \Bigl(\lim\sup_{t\uparrow\infty} \exp\{-\lambda t\}X_t(B)=\infty\Bigr)>0, \quad\text{and}\quad P_\mu \Bigl(\lim\sup_{t\uparrow \infty} \exp\{-\lambda_ct\} X_t(B)<\infty\Bigr)=1. \] The method of proof is probabilistic, using ``spine'' or ``immortal particle'' decomposition.
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 2100712
zbMATH Keywords: branching diffusion
zbMATH Keywords: superprocess
zbMATH Keywords: spine decomposition
zbMATH Keywords: immortal particle decomposition
reviewed by: Heinrich Hering
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1078415829
OpenAlex ID: W2062926417
cites work: Change of measures for Markov chains and the \(L\log L\) theorem for branching processes
cites work: Measure change in multitype branching
cites work: Q4003876
cites work: Q3142396
cites work: Probability
cites work: Superprocesses and partial differential equations
cites work: Branching exit Markov systems and superprocesses
cites work: On the construction and support properties of measure-valued diffusions on \(D\subseteq \mathbb{R}^d\) with spatially dependent branching
cites work: A scaling limit theorem for a class of superdiffusions
cites work: Q4514234
cites work: A decomposition of the (1 + β)-superprocess conditioned on survival
cites work: Two representations of a conditioned superprocess
cites work: Weighted Occupation Time for Branching Particle Systems and a Representation for the Supercritical Superprocess
cites work: The Galton-Watson tree conditioned on its height
cites work: On the supports of measure-valued critical branching Brownian motion
cites work: Q3774629
cites work: Stability of Critical Cluster Fields
cites work: Q5284193
cites work: Conceptual proofs of \(L\log L\) criteria for mean behavior of branching processes
cites work: A limit theorem for particle numbers in bounded domains of a branching diffusion process
cites work: Conditioned super-Brownian motion
cites work: Q4318497
cites work: Transience, recurrence and local extinction properties of the support for supercritical finite measure-valued diffusions
cites work: Q4734585
cites work: On the conditioned exit measures of super Brownian motion
cites work: Q4937701
DOI: 10.1214/AOP/1078415829
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description: scientific article; zbMATH DE number 2100712