The Annals of Probability

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

Donald A. Dawson and Edwin A. Perkins

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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

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855746

Digital Object Identifier
doi:10.1214/aop/1022855746

Mathematical Reviews number (MathSciNet)
MR1634416

Zentralblatt MATH identifier
0938.60042

Subjects
Primary: 60G57: Random measures 60H15: Stochastic partial differential equations [See also 35R60] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Measure-valued processes branching superprocesses equilibrium distribution dual process coexistence of types

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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