Occupation time fluctuations of Poisson and equilibrium branching systems in critical and large dimensions

TL;DR

This paper establishes limit theorems for the rescaled occupation time fluctuations of critical branching particle systems with symmetric stable motion in high dimensions, showing convergence to generalized Wiener processes.

Contribution

It provides new limit theorems for occupation time fluctuations in critical and large dimensions, extending understanding of branching systems with stable motion.

Findings

Convergence to generalized Wiener processes in high dimensions.

Functional convergence under additional branching law assumptions.

Results applicable to systems starting from Poisson or equilibrium distributions.

Abstract

Limit theorems are presented for the rescaled occupation time fluctuation process of a critical finite variance branching particle system in $\mathbb{R}^{d}$ with symmetric $\alpha$-stable motion starting off from either a standard Poisson random field or the equilibrium distribution for critical $d=2\alpha$ and large $d>2\alpha$ dimensions. The limit processes are generalised Wiener processes. The obtained convergence is in space-time, finite-dimensional distributions sense. With the addtional assumption on the branching law we obtain functional convergence.