Occupation times of subcritical branching immigration systems with Markov motions

Abstract

We consider a branching system consisting of particles moving according to a Markov family in R d and undergoing subcritical branching with a constant rate V > 0 . New particles immigrate to the system according to a homogeneous space–time Poisson random field. The process of the fluctuations of the rescaled occupation time is studied with very mild assumptions on the Markov family. In this general setting a functional central limit theorem is proved. The subcriticality of the branching law is crucial for the limit behaviour and in a sense overwhelms the properties of the particles’ motion. It is for this reason that the limit is the same for all dimensions and can be obtained for a wide class of Markov processes. Another consequence is the form of the limit —an S ′ ( R d ) -valued Wiener process with a simple temporal structure and a complicated spatial one. This behaviour contrasts sharply with the case of critical branching systems.