Occupation time fluctuations of an infinite-variance branching system in large dimensions

Abstract

We prove limit theorems for rescaled occupation time fluctuations of a $(d,α,β)$-branching particle system (particles moving in $ℝ^d$ according to a spherically symmetric $α$-stable Lévy process, $(1+β)$-branching, $0<β<1$, uniform Poisson initial state), in the cases of critical dimension, $d=α(1+β)/β$, and large dimensions, $d>α(1+β)/β$. The fluctuation processes are continuous but their limits are stable processes with independent increments, which have jumps. The convergence is in the sense of finite-dimensional distributions, and also of space-time random fields (tightness does not hold in the usual Skorohod topology). The results are in sharp contrast with those for intermediate dimensions, $α/β<d<d(1+β)/β$, where the limit process is continuous and has long-range dependence. The limit process is measure-valued for the critical dimension, and $\mathcal{S}^\prime(\mathbb{R}^{d})$-valued for large dimensions. We also raise some questions of interpretation of the different types of dimension-dependent results obtained in the present and previous papers in terms of properties of the particle system.