Diffusion approximations of age-and-position dependent branching processes

By a (G, F, h) age-and-position dependent branching process we mean a process in which individuals reproduce according to an age dependent branching process with age distribution function G(t) and offspring distribution generating function F, the individuals (located in RN) can not move and the distance of a new individual from its parent is governed by a probability density function h(r). For each positive integer n, let Zn(t,dx) be the number of individuals in dx at time t of the (G, Fn,hn) age-and-position dependent branching process. It is shown that under appropriate conditions on G, Fn and hn, the finite dimensional distribution of Zn(nt, dx)/n converges, as n --> [infinity], to the corresponding law of a diffusion continuous state branching process X(t,dx) determined by a [psi]-semigroup {[psi]t: t [greater-or-equal, slanted] 0}. The [psi]-semigroup {[psi]t} is the solution of a non-linear evolution equation. A semigroup convergence theorem due to Kurtz [10], which gives conditions for convergence in distribution of a sequence of non-Markovian processes to a Markov process, provides the main tools.