Longtime behavior of completely positively correlated Symbiotic Branching Model

TL;DR

This paper analyzes the long-term dynamics of a continuous-state Symbiotic Branching Model with perfect positive correlation, revealing detailed behavior in the transient regime for the case where the correlation parameter equals one.

Contribution

It provides a precise description of the longtime behavior of the SBM with correlation one, filling a key gap in understanding for the transient regime.

Findings

Behavior characterized for $ ho=1$ in the transient regime.

Identifies conditions for coexistence and non-coexistence.

Extends understanding of SBM dynamics with positive correlation.

Abstract

We study the longtime behavior of a continuous state Symbiotic Branching Model (SBM). SBM can be seen as a unified model generalizing the Stepping Stone Model, Mutually Catalytic Branching Processes, and the Parabolic Anderson Model. It was introduced by Etheridge and Fleischmann in 2004. The key parameter in these models is the local correlation $\rho$ between the driving Brownian Motions. The longtime behavior of all SBM exhibits a dichotomy between coexistence and non-coexistence of the two populations depending on the recurrence and transience of the migration and also in many cases on the branching rate. The most significant gap in the understanding of the longtime behavior of SBM is for positive correlations in the transient regime. In this article we give a precise description of the longtime behavior of the SBM with $\rho=1$ with not necessarily identical initial conditions.