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

Abstract

The boundary behavior of continuous-state branching processes with quadratic\ncompetition is studied in whole generality. We first observe that despite\ncompetition, explosion can occur for certain branching mechanisms. We obtain a\nnecessary and sufficient condition for $\\infty$ to be accessible in terms of\nthe branching mechanism and the competition parameter $c>0$. We show that when\n$\\infty$ is inaccessible, it is always an entrance boundary. In the case where\n$\\infty$ is accessible, explosion can occur either by a single jump to $\\infty$\n(the process at $z$ jumps to $\\infty$ at rate $\\lambda z$ for some $\\lambda>0$)\nor by accumulation of large jumps over finite intervals. We construct a natural\nextension of the minimal process and show that when $\\infty$ is accessible and\n$0\\leq \\frac{2\\lambda}{c}<1$, the extended process is reflected at $\\infty$. In\nthe case $\\frac{2\\lambda}{c}\\geq 1$, $\\infty$ is an exit of the extended\nprocess. When the branching mechanism is not the Laplace exponent of a\nsubordinator, we show that the process with reflection at $\\infty$ get extinct\nalmost-surely. Moreover absorption at $0$ is almost-sure if and only if Grey's\ncondition is satisfied. When the branching mechanism is the Laplace exponent of\na subordinator, necessary and sufficient conditions are given for a stationary\ndistribution to exist. The Laplace transform of the latter is provided. The\nstudy is based on classical time-change arguments and on a new duality method\nrelating logistic CSBPs with certain generalized Feller diffusions.\n