2017/11/18 by Foucart, Clément · 2 citations
#60J70 #60J80 #92D25 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1711.06827
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 ∞ to be accessible in terms of the branching mechanism and the competition parameter c>0. We show that when ∞ is inaccessible, it is always an entrance boundary. In the case where ∞ is accessible, explosion can occur either by a single jump to ∞ (the process at z jumps to ∞ at rate λz for some λ>0) or by accumulation of large jumps over finite intervals. We construct a natural extension of the minimal process and show that when ∞ is accessible and 0≤ (2λ)/(c)<1, the extended process is reflected at ∞. In the case (2λ)/(c)≥ 1, ∞ 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 ∞ 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.