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April 2000 Finite time extinction of superprocesses with catalysts
Donald A. Dawson, Klaus Fleischmann, Carl Mueller
Ann. Probab. 28(2): 603-642 (April 2000). DOI: 10.1214/aop/1019160254

Abstract

Consider a catalytic super-Brownian motion $X =X^\Gamma$ with finite variance branching. Here “catalytic ” means that branching of the reactant $X$ is only possible in the presence of some catalyst. Our intrinsic example of a catalyst is a stable random measure $\Gamma$ on $\mathsf{R}$ of index $0 <\gamma<1$. Consequently, here the catalyst is located in a countable dense subset of $\mathsf{R}$. Starting with a finite reactant mass $X_0$ supported by a compact set, $X$ is shown to die in finite time.We also deal with two other cases, with a power low catalyst and with a super-random walk on $\mathsf{Z^d}$ withan i.i.d.catalyst.

Our probabilistic argument uses the idea of good and bad historical paths of reactant “particles ”during time periods $[T_n, T_{n +1}$. Good paths have a signi .cant collision local time with the catalyst, and extinction can be shown by individual time change according to the collision local time and a comparison with Feller’s branching diffusion. On the other hand, the remaining bad paths are shown to have a small expected mass at time $T_{n +1}$ which can be controlled by the hitting probability of point catalysts and the collision local time spent on them.

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Donald A. Dawson. Klaus Fleischmann. Carl Mueller. "Finite time extinction of superprocesses with catalysts." Ann. Probab. 28 (2) 603 - 642, April 2000. https://doi.org/10.1214/aop/1019160254

Information

Published: April 2000
First available in Project Euclid: 18 April 2002

zbMATH: 1044.60073
MathSciNet: MR1782268
Digital Object Identifier: 10.1214/aop/1019160254

Subjects:
Primary: 60J80
Secondary: 60G57 , 60J55

Keywords: branching rate functional , Catalytic super-Brownian motion , Collision local time , Comparison , critical branching , finite time extinction , finite time survival , good and bad paths , historical superprocess , interacting Feller’s branching diffusion , measure-valued branching , random medium , stable catalyst. , stopped historical superprocess , stopped measures , super-random walk

Rights: Copyright © 2000 Institute of Mathematical Statistics

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Vol.28 • No. 2 • April 2000
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