The Annals of Probability
- Ann. Probab.
- Volume 38, Number 2 (2010), 479-497.
A Trotter-type approach to infinite rate mutually catalytic branching
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.
Ann. Probab., Volume 38, Number 2 (2010), 479-497.
First available in Project Euclid: 9 March 2010
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Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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