Infinite rate mutually catalytic branching

TL;DR

This paper studies the limit of a mutually catalytic branching process as the branching rate becomes infinite, showing convergence to a discontinuous process and providing multiple characterizations.

Contribution

It introduces a new limiting process for infinite rate mutually catalytic branching and offers detailed descriptions and a strong construction method.

Findings

Convergence of the process as the rate approaches infinity

Description of the limiting process via semigroup and martingale problem

Path properties inferred from planar Brownian motion

Abstract

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.