## The Annals of Probability

- Ann. Probab.
- Volume 38, Number 4 (2010), 1690-1716.

### Infinite rate mutually catalytic branching

Achim Klenke and Leonid Mytnik

#### Abstract

Consider the mutually catalytic branching process with finite branching rate *γ*. We show that as *γ* → ∞, this process converges in finite-dimensional distributions (in time) to a certain discontinuous process. We give descriptions of this process in terms of its semigroup in terms of the infinitesimal generator and as the solution of a martingale problem. We also give a strong construction in terms of a planar Brownian motion from which we infer a path property of the process.

This is the first paper in a series or three, wherein we also construct an interacting version of this process and study its long-time behavior.

#### Article information

**Source**

Ann. Probab., Volume 38, Number 4 (2010), 1690-1716.

**Dates**

First available in Project Euclid: 8 July 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1278593965

**Digital Object Identifier**

doi:10.1214/09-AOP520

**Mathematical Reviews number (MathSciNet)**

MR2663642

**Zentralblatt MATH identifier**

1204.60081

**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 strong construction stochastic differential equations

#### Citation

Klenke, Achim; Mytnik, Leonid. Infinite rate mutually catalytic branching. Ann. Probab. 38 (2010), no. 4, 1690--1716. doi:10.1214/09-AOP520. https://projecteuclid.org/euclid.aop/1278593965