## The Annals of Probability

### Long-time behavior and coexistence in a mutually catalytic branching model

#### Abstract

We study a system of two interacting populations which undergo random migration and mutually catalytic branching. The branching rate of one population at a site is proportional to the mass of the other population at the site. The system is modelled by an infinite system of stochastic differential equations, allowing symmetric Markov migration, if the set of sites is discrete $(\mathbb{Z}^d)$, or by a stochastic partial differential equation with Brownian migration if the set of sites is the real line. A duality technique of Leonid Mytnik, which gives uniqueness in law, is used to examine the long-time behavior of the solutions. For example, with uniform initial conditions, the process converges to an equilibrium distribution as $t \to \infty$, and there is coexistence of types in the equilibrium “iff ” the random migration is transient.

#### Article information

Source
Ann. Probab., Volume 26, Number 3 (1998), 1088-1138.

Dates
First available in Project Euclid: 31 May 2002

https://projecteuclid.org/euclid.aop/1022855746

Digital Object Identifier
doi:10.1214/aop/1022855746

Mathematical Reviews number (MathSciNet)
MR1634416

Zentralblatt MATH identifier
0938.60042

#### Citation

Dawson, Donald A.; Perkins, Edwin A. Long-time behavior and coexistence in a mutually catalytic branching model. Ann. Probab. 26 (1998), no. 3, 1088--1138. doi:10.1214/aop/1022855746. https://projecteuclid.org/euclid.aop/1022855746

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