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Long-time behavior and coexistence in a mutually catalytic branching model. (English) Zbl 0938.60042

Authors’ 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.
Reviewer: K.Fleischmann

MSC:

60G57 Random measures
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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