Electronic Journal of Probability
- Electron. J. Probab.
- Volume 8 (2003), paper no. 8, 59 p.
Competing Species Superprocesses with Infinite Variance
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.
Electron. J. Probab., Volume 8 (2003), paper no. 8, 59 p.
First available in Project Euclid: 23 May 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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.)
Superprocess with killing competing superprocesses interactive superprocesses superprocess with immigration measure-valued branching interactive branching state-dependent branching collision measure collision local time martingale problem
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