Continuous-state branching processes with competition: duality and reflection at Infinity

TL;DR

This paper investigates the boundary behavior of continuous-state branching processes with competition, identifying conditions for explosion, reflection, and extinction, and introduces a duality method to analyze these processes comprehensively.

Contribution

It provides a complete characterization of boundary behaviors, including explosion and reflection, for logistic CSBPs and introduces a novel duality approach with generalized Feller diffusions.

Findings

Explosion can occur despite competition under certain conditions.

When the boundary at infinity is accessible, the process can be reflected or absorbed depending on parameters.

Conditions for the existence of stationary distributions are established for specific branching mechanisms.

Abstract

The boundary behavior of continuous-state branching processes with quadratic competition is studied in whole generality. We first observe that despite competition, explosion can occur for certain branching mechanisms. We obtain a necessary and sufficient condition for $\infty$ to be accessible in terms of the branching mechanism and the competition parameter $c>0$. We show that when $\infty$ is inaccessible, it is always an entrance boundary. In the case where $\infty$ is accessible, explosion can occur either by a single jump to $\infty$ (the process at $z$ jumps to $\infty$ at rate $\lambda z$ for some $\lambda>0$) or by accumulation of large jumps over finite intervals. We construct a natural extension of the minimal process and show that when $\infty$ is accessible and $0\leq \frac{2\lambda}{c}<1$, the extended process is reflected at $\infty$. In the case $\frac{2\lambda}{c}\geq 1$, $\infty$ is an exit of the extended process. When the branching mechanism is not the Laplace exponent of a subordinator, we show that the process with reflection at $\infty$ get extinct almost-surely. Moreover absorption at $0$ is almost-sure if and only if Grey's condition is satisfied. When the branching mechanism is the Laplace exponent of a subordinator, necessary and sufficient conditions are given for a stationary distribution to exist. The Laplace transform of the latter is provided. The study is based on classical time-change arguments and on a new duality method relating logistic CSBPs with certain generalized Feller diffusions.