Competing species superprocesses with infinite variance
- Fleischmann, Klaus
- Mytnik, Leonid
2010 Mathematics Subject Classification
- 60K35 60G57 60J80
- 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
We study pairs of interacting measure-valued branching processes (superprocesses) with $(1+beta)$-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 fact that 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.
- Electr. J. Probab. , vol. 8, no. 8, 59 pp