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

TL;DR

This paper constructs and characterizes an infinite rate mutually catalytic branching process on countable sites, showing it as a limit of finite rate models and providing multiple characterizations including martingale problems and stochastic integral equations.

Contribution

It introduces a new infinite rate mutually catalytic branching process, extending the finite rate model and establishing its uniqueness and convergence properties.

Findings

The process is the limit of finite rate models as the rate goes to infinity.

Unique solution characterized by a martingale problem.

Representation as a weak solution of stochastic integral equations.

Abstract

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.