Occupation time fluctuation limits of infinite variance equilibrium branching systems

TL;DR

This paper proves limit theorems for the fluctuations of occupation times in a stable branching particle system with infinite variance, revealing different behaviors across dimensions and identifying a fractional stable motion in intermediate dimensions.

Contribution

It introduces new limit theorems for occupation time fluctuations in infinite variance branching systems, including the derivation of a fractional stable motion in intermediate dimensions.

Findings

In intermediate dimensions, the limit process is a fractional stable motion.

In critical and large dimensions, the limit processes have independent increments.

The study characterizes the long-range dependence structure of the limit processes.

Abstract

We establish limit theorems for the fluctuations of the rescaled occupation time of a $(d,\alpha,\beta)$-branching particle system. It consists of particles moving according to a symmetric $\alpha$-stable motion in $\mathbb{R}^d$. The branching law is in the domain of attraction of a (1+$\beta$)-stable law and the initial condition is an equilibrium random measure for the system (defined below). In the paper we treat separately the cases of intermediate $\alpha/\beta<d<(1+\beta)\alpha/\beta$, critical $d=(1+\beta)\alpha/\beta$ and large $d>(1+\beta)\alpha/\beta $ dimensions. In the most interesting case of intermediate dimensions we obtain a version of a fractional stable motion. The long-range dependence structure of this process is also studied. Contrary to this case, limit processes in critical and large dimensions have independent increments.