Infinite rate mutually catalytic branching

Details Infinite rate mutually catalytic branching Infinite rate mutually catalytic branching

2008-09-26

arxiv.org/abs/0809.4554v2

Annals of Probability 2010, Vol. 38, No. 4, 1690-1716

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/09-AOP520 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/09-AOP520

Consider the mutually catalytic branching process with finite branching rate $\gamma$. We show that as $\gamma\to\infty$, this process converges in finite-dimensional distributions (in time) to a certain discontinuous process. We give descriptions of this process in terms of its semigroup in terms of the infinitesimal generator and as the solution of a martingale problem. We also give a strong construction in terms of a planar Brownian motion from which we infer a path property of the process. This is the first paper in a series or three, wherein we also construct an interacting version of this process and study its long-time behavior.

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