Electronic Journal of Probability

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

Clément Foucart

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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 $ gets 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.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 33, 38 pp.

Dates
Received: 26 September 2018
Accepted: 24 March 2019
First available in Project Euclid: 9 April 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1554775412

Digital Object Identifier
doi:10.1214/19-EJP299

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 92D25: Population dynamics (general)

Keywords
continuous-state branching process generalized Feller diffusion branching process with interaction explosion coming down from infinity entrance boundary reflecting boundary Lamperti’s time change duality

Rights
Creative Commons Attribution 4.0 International License.

Citation

Foucart, Clément. Continuous-state branching processes with competition: duality and reflection at infinity. Electron. J. Probab. 24 (2019), paper no. 33, 38 pp. doi:10.1214/19-EJP299. https://projecteuclid.org/euclid.ejp/1554775412


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