The Annals of Probability

Finite time extinction of superprocesses with catalysts

Donald A. Dawson, Klaus Fleischmann, and Carl Mueller

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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.

Article information

Source
Ann. Probab., Volume 28, Number 2 (2000), 603-642.

Dates
First available in Project Euclid: 18 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1019160254

Digital Object Identifier
doi:10.1214/aop/1019160254

Mathematical Reviews number (MathSciNet)
MR1782268

Zentralblatt MATH identifier
1044.60073

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J55: Local time and additive functionals 60G57: Random measures

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

Citation

Dawson, Donald A.; Fleischmann, Klaus; Mueller, Carl. Finite time extinction of superprocesses with catalysts. Ann. Probab. 28 (2000), no. 2, 603--642. doi:10.1214/aop/1019160254. https://projecteuclid.org/euclid.aop/1019160254


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