Occupation times of subcritical branching immigration system with Markov motions

TL;DR

This paper proves a functional central limit theorem for the occupation time fluctuations of a subcritical branching system with particles moving via Markov processes and immigrating according to a Poisson field, revealing universal limit behavior.

Contribution

It establishes a general functional CLT for subcritical branching systems with Markov motions under mild assumptions, showing the limit is a spatially complex Wiener process independent of dimension.

Findings

Limit process is a spatially complex Wiener process.

Limit behavior is universal across dimensions due to subcriticality.

The result contrasts with critical branching systems.

Abstract

We consider a branching system consisting of particles moving according to a Markov family in $\Rd$ and undergoing subcritical branching with a constant rate $V>0$. New particles immigrate to the system according to 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 - $\SP$-valued Wiener process with a simple temporal structure and a complicated spatial one. This behaviour contrasts sharply with the case of critical branching systems.