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July 1998 Long-time behavior and coexistence in a mutually catalytic branching model
Donald A. Dawson, Edwin A. Perkins
Ann. Probab. 26(3): 1088-1138 (July 1998). DOI: 10.1214/aop/1022855746


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.


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Donald A. Dawson. Edwin A. Perkins. "Long-time behavior and coexistence in a mutually catalytic branching model." Ann. Probab. 26 (3) 1088 - 1138, July 1998.


Published: July 1998
First available in Project Euclid: 31 May 2002

zbMATH: 0938.60042
MathSciNet: MR1634416
Digital Object Identifier: 10.1214/aop/1022855746

Primary: 60G57 , 60H15 , 60J80
Secondary: 60H10 , 60J60 , 60K35

Keywords: branching , coexistence of types , Dual process , equilibrium distribution , Measure-valued processes , Superprocesses

Rights: Copyright © 1998 Institute of Mathematical Statistics

Vol.26 • No. 3 • July 1998
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