Boundary behavior of continuous-state interacting multi-type branching processes with immigration

TL;DR

This paper analyzes the boundary behavior of continuous-state multi-type branching processes with immigration, establishing conditions for boundary hitting and non-hitting in various stochastic settings.

Contribution

It provides new sufficient conditions for boundary non-attainment and boundary hitting in CIMBI processes, considering both diffusion and jump cases.

Findings

Processes with small immigration and diffusion noise can hit the boundary with positive probability.

Under certain conditions, processes with jumps of finite activity hit the boundary with positive probability.

Processes can be shown to never hit the boundary when starting from the interior under specific conditions.

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.