The Annals of Probability

A Trotter-type approach to infinite rate mutually catalytic branching

Achim Klenke and Mario Oeler

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Abstract

Dawson and Perkins [Ann. Probab. 26 (1988) 1088–1138] constructed a stochastic model of an interacting two-type population indexed by a countable site space which locally undergoes a mutually catalytic branching mechanism. In Klenke and Mytnik [Preprint (2008), arXiv:0901.0623], it is shown that as the branching rate approaches infinity, the process converges to a process that is called the infinite rate mutually catalytic branching process (IMUB). It is most conveniently characterized as the solution of a certain martingale problem. While in the latter reference, a noise equation approach is used in order to construct a solution to this martingale problem, the aim of this paper is to provide a Trotter-type construction.

The construction presented here will be used in a forthcoming paper, Klenke and Mytnik [Preprint (2009)], to investigate the long-time behavior of IMUB (coexistence versus segregation of types).

This paper is partly based on the Ph.D. thesis of the second author (2008), where the Trotter approach was first introduced.

Article information

Source
Ann. Probab., Volume 38, Number 2 (2010), 479-497.

Dates
First available in Project Euclid: 9 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1268143524

Digital Object Identifier
doi:10.1214/09-AOP488

Mathematical Reviews number (MathSciNet)
MR2642883

Zentralblatt MATH identifier
1191.60112

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J65: Brownian motion [See also 58J65] 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]

Keywords
Mutually catalytic branching martingale problem stochastic differential equations population dynamics Trotter formula

Citation

Klenke, Achim; Oeler, Mario. A Trotter-type approach to infinite rate mutually catalytic branching. Ann. Probab. 38 (2010), no. 2, 479--497. doi:10.1214/09-AOP488. https://projecteuclid.org/euclid.aop/1268143524


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References

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