The Annals of Probability

Mutually catalytic branching in the plane: Finite measure states

Donald A. Dawson, Alison M. Etheridge, Klaus Fleischmann, Leonid Mytnik, Edwin A. Perkins, and Jie Xiong

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Abstract

We study a pair of populations in $\mathbb{R}^{2}$ which undergo diffusion and branching. The system is interactive in that the branching rate of each type is proportional to the local density of the other type. For a diffusion rate sufficiently large compared with the branching rate, the model is constructed as the unique pair of finite measure-valued processes which satisfy a martingale problem involving the collision local time of the solutions. The processes are shown to have densities at fixed times which live on disjoint sets and explode as they approach the interface of the two populations. In the long-term limit, global extinction of one type is shown. The process constructed is a rescaled limit of the corresponding $\mathbb{Z}^{2}$-lattice model studied by D. A. Dawson and E. A. Perkins and resolves the large scale mass-time-space behavior of that model.

Article information

Source
Ann. Probab., Volume 30, Number 4 (2002), 1681-1762.

Dates
First available in Project Euclid: 10 December 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1039548370

Digital Object Identifier
doi:10.1214/aop/1039548370

Mathematical Reviews number (MathSciNet)
MR1944004

Zentralblatt MATH identifier
1017.60098

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60G57: Random measures 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Catalytic super-Brownian motion catalytic super-random walk collision local time duality superprocesses martingale problem segregation of types stochastic PDE

Citation

Dawson, Donald A.; Etheridge, Alison M.; Fleischmann, Klaus; Mytnik, Leonid; Perkins, Edwin A.; Xiong, Jie. Mutually catalytic branching in the plane: Finite measure states. Ann. Probab. 30 (2002), no. 4, 1681--1762. doi:10.1214/aop/1039548370. https://projecteuclid.org/euclid.aop/1039548370


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  • KNOXVILLE, TENNESSEE 37996-1300 E-MAIL: jxiong@math.utk.edu

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