## The Annals of Probability

### Mutually catalytic branching in the plane: Finite measure states

#### 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

#### 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