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

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.

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