## Electronic Journal of Probability

### Competing Species Superprocesses with Infinite Variance

#### Abstract

We study pairs of interacting measure-valued branching processes (superprocesses) with alpha-stable migration and $(1+\beta)$-branching mechanism. The interaction is realized via some killing procedure. The collision local time for such processes is constructed as a limit of approximating collision local times. For certain dimensions this convergence holds uniformly over all pairs of such interacting superprocesses. We use this uniformity to prove existence of a solution to a competing species martingale problem under a natural dimension restriction. The competing species model describes the evolution of two populations where individuals of different types may kill each other if they collide. In the case of Brownian migration and finite variance branching, the model was introduced by Evans and Perkins (1994). The fact that now the branching mechanism does not have finite variance requires the development of new methods for handling the collision local time which we believe are of some independent interest.

#### Article information

Source
Electron. J. Probab., Volume 8 (2003), paper no. 8, 59 p.

Dates
First available in Project Euclid: 23 May 2016

https://projecteuclid.org/euclid.ejp/1464037581

Digital Object Identifier
doi:10.1214/EJP.v8-136

Mathematical Reviews number (MathSciNet)
MR1986840

Zentralblatt MATH identifier
1065.60145

#### Citation

Fleischmann, Klaus; Mytnik, Leonid. Competing Species Superprocesses with Infinite Variance. Electron. J. Probab. 8 (2003), paper no. 8, 59 p. doi:10.1214/EJP.v8-136. https://projecteuclid.org/euclid.ejp/1464037581

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