Longtime Behavior for Mutually Catalytic Branching with Negative Correlations

TL;DR

This paper establishes a precise dichotomy between extinction and coexistence for a two-type mutually catalytic branching model with negative correlations, extending previous results to infinite branching rates using moment bounds and second moment calculations.

Contribution

It introduces a new approach to analyze negatively correlated models, providing a detailed dichotomy proof that applies even at infinite branching rates.

Findings

Proves a dichotomy between extinction and coexistence.

Extends results to models with infinite branching rates.

Uses moment bounds and explicit second moment calculations.

Abstract

In several examples, dualities for interacting diffusion and particle systems permit the study of the longtime behavior of solutions. A particularly difficult model in which many techniques collapse is a two-type model with mutually catalytic interaction introduced by Dawson/Perkins for which they proved under some assumptions a dichotomy between extinction and coexistence directly from the defining equations. In the present article we show how to prove a precise dichotomy for a related model with negatively correlated noises. The proof uses moment bounds on exit-times of correlated Brownian motions from the first quadrant and explicit second moment calculations. Since the uniform integrability bound is independent of the branching rate our proof can be extended to infinite branching rate processes.