Boundary behavior of continuous-state interacting multi-type branching processes with immigration
Peng Jin, Jiaqi Zhou
TL;DR
This work investigates the boundary behavior of continuous-state interacting multi-type branching processes with immigration (CIMBI) by constructing a strong SDE-based model in $\mathbb{R}_{+}^{d}$ with an interaction matrix $c$ (where $c_{ii}<0$) and analyzing when the process can hit or avoid the boundary. It develops a Foster-Lyapunov framework to establish sufficient conditions for boundary non-attainment and employs a diffusion-comparison/Harris-recurrence approach to obtain boundary attainment results, including extensions to finite-activity jumps. The results show that, under diffusion with small immigration, boundary attainment can occur almost surely under certain negativity conditions on the interaction structure, while with finite activity jumps hitting occurs with positive probability; for competitive regimes, these conditions can be further relaxed. Collectively, the paper clarifies extinction and boundary-hitting mechanisms in CIMBI models, aiding theoretical understanding and practical modeling of multi-type populations with immigration and inter-type interactions.
Abstract
In this paper, we study continuous-state interacting multi-type branching processes with immigration (CIMBI processes), where inter-specific interactions -- whether competitive, cooperative, or of a mixed type -- are proportional to the product of their type-population masses. We establish sufficient conditions for the CIMBI process to never hit the boundary $\partial\mathbb{R}_{+}^{d}$ when starting from the interior of $\mathbb{R}_{+}^{d}$. Additionally, we present two results concerning boundary attainment. In the first, we consider the diffusion case and prove that when the constant immigration rate is small and diffusion noise is present in each direction, the CIMBI process will almost surely hit the boundary $\partial\mathbb{R}_{+}^{d}$. In the second result, under similar conditions on the constant immigration rate and diffusion noise, but with jumps of finite activity, we show that the CIMBI process hits the boundary $\partial\mathbb{R}_{+}^{d}$ with positive probability.