Space-time harmonic functions and age-dependent branching processes

Abstract

LetXbe an age-dependent branching process with lifetime distributionGand age-dependent generating function π(y,s) = σk= 0∞pk(y)sk. We assume thatGis right-continuous andG(0+) =G(0) = 0. The base state spaceSis [0,T) whereT= inf{t:G(t) = 1}. Setm(y) = σk= 0∞k pk(y) andThen extinction occurs with probability one iffm≤ 1. In the case wherem> 1, define the Malthusian parameter λ to be the unique (positive) root ofand setonS.is a-space-time harmonic function of the processXand the corresponding non-negative martingaleconverges w.p.l to a random variableW; furthermore, under a regularity assumption,Wis non-trivial iffwhereandIf 0 <a≤ Φ ≤ β < ∞, for some constantsa, β, thenw.p.l, whereZtis the number of particles at timet.