2025/04/07 by Eran Avneri, Avneri, Eran, Leonid Mytnik +1
Mathematics · Medicine · #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2504.04792
openalex publication_date 2025/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The symbiotic branching model in ℝ describes the behavior of two branching populations migrating in space ℝ in terms of a corresponding system of stochastic partial differential equations. The system is parametrized with a correlation parameter ρ, 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 ρ∈ [-1,1), weak uniqueness for the completely positively correlated case of ρ= 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.