Uniqueness and Longtime Behavior of the Completely Positively Correlated Symbiotic Branching Model

The symbiotic branching model in $\mathbb{R}$ describes the behavior of two branching populations migrating in space $\mathbb{R}$ in terms of a corresponding system of stochastic partial differential equations. The system is parametrized with a correlation parameter $\rho$, which takes values in $[-1,1]$ and governs the correlation between the branching mechanisms of the two populations. While existence and uniqueness for this system were established for $\rho \in [-1,1)$, weak uniqueness for the completely positively correlated case of $\rho = 1$ has been an open problem. In this paper, we resolve this problem, establishing weak uniqueness for the corresponding system of stochastic partial differential equations. The proof uses a new duality between the symbiotic branching model and the well-known parabolic Anderson model. Furthermore, we use this duality to investigate the long-term behavior of the completely positively correlated symbiotic branching model. We show that, under suitable initial conditions, after a long time, one of the populations dies out. We treat the case of integrable initial conditions and the case of bounded non-integrable initial conditions with well-defined mean.