Consider the mutually catalytic branching process with finite branching rate γ. We show that as γ → ∞, 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.
"Infinite rate mutually catalytic branching." Ann. Probab. 38 (4) 1690 - 1716, July 2010. https://doi.org/10.1214/09-AOP520