Electronic Journal of Probability

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

Clément Foucart

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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

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

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

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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)

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

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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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