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 talk, we solve 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.