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In this paper we consider two related stochastic models. The first one is abranching system consisting of particles moving according to a Markov family inR^d and undergoing subcritical branching with a constant rate of V>0. Newparticles immigrate to the system according to a homogeneous space time Poissonrandom field. The second model is the superprocess corresponding to thebranching particle system. We study rescaled occupation time process and theprocess of its fluctuations with very mild assumptions on the Markov family. Inthe general setting a functional central limit theorem as well as large andmoderate deviations principles are proved. The subcriticality of the branchinglaw determines the behaviour in large time scales and in "overwhelms" theproperties of the particles' motion. For this reason the results are the samefor all dimensions and can be obtained for a wide class of Markov processes(both properties are unusual for systems with critical branching).

Occupation times of subcritical branching immigration systems with Markov motion, clt and deviations principles