Extinction properties of super-Brownian motions with additional spatially dependent mass production.
type: Item
instance of: scholarly article
title: Extinction properties of super-Brownian motions with additional spatially dependent mass production.
zbMATH Open document ID: 1046.60075
DOI: 10.1016/S0304-4149(99)00118-0
author: János Engländer
author: Klaus Fleischmann
published in: Stochastic Processes and their Applications
publication date: 2004-09-07T00:00:00Z
review text: Extinction and local extinction of the finite measure-valued continuous super-Brownian motion \(X\) on \(\mathbb R^d\) corresponding to the log-Laplace equation \(\frac {\partial }{\partial t} u =\frac {1}{2}\Delta u +\beta u -u^2\) are studied, where the additional mass production coefficient \(\beta \) is Hölder continuous and bounded from above. A threshold decay rate \(k_d| x| ^{-2}\) as \(| x| \to \infty \) is found such that the extinction does not occur if \(\beta \) is above the threshold, whereas it does occur below the threshold, where \(k_d=\frac {1}{8} (d-2)^2\) if \(d\leq 6\) and \(k_d=d-4\) if \(d>6\). An analogous criterion has been proved for the local extinction. It is also shown that in dimension 1 if \(\beta \) is not bounded from above, \(X\) does not exhibit local extinction and degenerates to a single point source \(\delta _0\).
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60J60
zbMATH DE Number: 2098322
zbMATH Keywords: measure-valued process
zbMATH Keywords: superprocess
zbMATH Keywords: extinction
zbMATH Keywords: branching
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2098322