## The Annals of Probability

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

### A Trotter-type approach to infinite rate mutually catalytic branching

#### 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