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

TL;DR

This paper analyzes occupation times in subcritical branching systems with Markov motion, establishing CLT and deviations principles, showing that subcriticality dominates particle motion effects across all dimensions.

Contribution

It provides the first general CLT and deviations principles for occupation times in subcritical branching systems with Markov motion under mild assumptions.

Findings

Functional CLT for occupation times

Large and moderate deviations principles established

Results hold for all dimensions and broad Markov processes

Abstract

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