Space-time harmonic functions and age-dependent branching processes

Abstract

Let X be an age-dependent branching process with lifetime distribution G and age-dependent generating function π(y,s) = σ_{k = 0}^∞p_k(y) s^k. We assume that G is right-continuous and G(0+) = G(0) = 0. The base state space S is [0,T) where T = inf{t : G(t) = 1}. Set m(y) = σ_{k = 0}^∞k p_k(y) and Then extinction occurs with probability one iff m ≤ 1. In the case where m > 1, define the Malthusian parameter λ to be the unique (positive) root of and set on S. is a -space-time harmonic function of the process X and the corresponding non-negative martingale converges w.p.l to a random variable W; furthermore, under a regularity assumption, W is non-trivial iff where and If 0 < a ≤ Φ ≤ β < ∞, for some constants a, β, then w.p.l, where Z_t is the number of particles at time t.