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October 2002 Mutually catalytic branching in the plane: Finite measure states
Donald A. Dawson, Alison M. Etheridge, Klaus Fleischmann, Leonid Mytnik, Edwin A. Perkins, Jie Xiong
Ann. Probab. 30(4): 1681-1762 (October 2002). DOI: 10.1214/aop/1039548370

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.

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Donald A. Dawson. Alison M. Etheridge. Klaus Fleischmann. Leonid Mytnik. Edwin A. Perkins. Jie Xiong. "Mutually catalytic branching in the plane: Finite measure states." Ann. Probab. 30 (4) 1681 - 1762, October 2002. https://doi.org/10.1214/aop/1039548370

Information

Published: October 2002
First available in Project Euclid: 10 December 2002

zbMATH: 1017.60098
MathSciNet: MR1944004
Digital Object Identifier: 10.1214/aop/1039548370

Subjects:
Primary: 60K35
Secondary: 60G57 , 60J80

Keywords: Catalytic super-Brownian motion , catalytic super-random walk , Collision local time , Duality , Martingale problem , segregation of types , Stochastic pde , Superprocesses

Rights: Copyright © 2002 Institute of Mathematical Statistics

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Vol.30 • No. 4 • October 2002
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