Infinite rate mutually catalytic branching in infinitely many colonies. Construction, characterization and convergence

Details Infinite rate mutually catalytic branching in infinitely many colonies. Construction, characterization and convergence Infinite rate mutually catalytic branching in infinitely many colonies. Construction, characterization and convergence

2009-01-06

arxiv.org/abs/0901.0623v3

Mathematics

Probability

35 pages

Scientific paper

We construct a mutually catalytic branching process on a countable site space with infinite "branching rate". The finite rate mutually catalytic model, in which the rate of branching of one population at a site is proportional to the mass of the other population at that site, was introduced by Dawson and Perkins in [DP98]. We show that our model is the limit for a class of models and in particular for the Dawson-Perkins model as the rate of branching goes to infinity. Our process is characterized as the unique solution to a martingale problem. We also give a characterization of the process as a weak solution of an infinite system of stochastic integral equations driven by a Poisson noise.

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