[Paper Review] Continuous-state branching processes with competition: duality and reflection at Infinity

Clément Foucart|arXiv (Cornell University)|Nov 18, 2017

Stochastic processes and statistical mechanics31 references18 citations

TL;DR

This paper studies continuous-state branching processes with quadratic competition, establishing a necessary and sufficient condition for infinity to be accessible or an entrance boundary. It introduces a novel duality between logistic CSBPs and generalized Feller diffusions, proving that when infinity is accessible, the process can be reflected at infinity under certain parameter conditions, with almost sure extinction at 0 if Grey's condition holds.

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.

Motivation & Objective

• To characterize the boundary behavior of continuous-state branching processes with quadratic competition.
• To determine when infinity is accessible or an entrance boundary, depending on the branching mechanism and competition parameter.
• To construct a natural extension of the minimal process when infinity is accessible, and analyze its reflection or exit behavior.
• To establish conditions for almost sure extinction at 0 and existence of a stationary distribution under different branching mechanisms.
• To develop a new duality method linking logistic CSBPs to generalized Feller diffusions for deeper structural analysis.

Proposed method

• Uses Lamperti's time-change technique to relate the logistic CSBP to a time-changed Feller diffusion.
• Applies a novel duality framework between logistic CSBPs and generalized Feller diffusions to analyze boundary behavior.
• Employs classical time-change arguments and Laplace transform techniques to study first entrance times and semigroup properties.
• Applies the monotone convergence theorem and Lebesgue's theorem to analyze convergence of integrals in the Laplace transform of the stationary distribution.
• Derives necessary and sufficient conditions for the existence of a stationary distribution via the Laplace transform of the limiting measure.
• Uses optional stopping and martingale arguments to compute the Laplace transform of the first hitting time of a level by the associated Ornstein-Uhlenbeck-type process.

Experimental results

Research questions

• RQ1 Under what conditions is infinity accessible in a continuous-state branching process with quadratic competition?
• RQ2 Can a logistic CSBP be reflected at infinity, and under what parameter constraints does this occur?
• RQ3 What determines whether the process almost surely goes extinct at 0, and how does this relate to Grey’s condition?
• RQ4 When does a stationary distribution exist for a logistic CSBP, and what is its Laplace transform?
• RQ5 How does the duality between logistic CSBPs and generalized Feller diffusions help characterize the boundary behavior at infinity?

Key findings

• Infinity is accessible if and only if the branching mechanism and competition parameter satisfy a specific integral condition involving the Laplace exponent.
• When infinity is inaccessible, it is always an entrance boundary, meaning the process can come down from infinity.
• If infinity is accessible and $ \frac{2\lambda}{c} < 1 $, the extended process is reflected at infinity; if $ \frac{2\lambda}{c} \geq 1 $, infinity is an exit boundary.
• Almost sure extinction at 0 occurs if and only if Grey’s condition is satisfied, regardless of the competition parameter.
• When the branching mechanism is the Laplace exponent of a subordinator, a stationary distribution exists if and only if a certain integral condition on the Lévy measure holds, and its Laplace transform is explicitly computed.
• The duality method enables a complete classification of boundary behavior, including the role of large jumps and explosion dynamics.