Occupation time fluctuations of an age-dependent critical binary branching particle system

Abstract

We study the limit fluctuations of the rescaled occupation time process of a branching particle system in $\mathbb{R}^d$, where the particles are subject to symmetric $\alpha$-stable migration ($0<\alpha\leq2$), critical binary branching, and general non-lattice lifetime distribution. We focus on two different regimes: lifetime distributions having finite expectation, and Pareto-type lifetime distributions, i.e. distributions belonging to the normal domain of attraction of a $\gamma$-stable law with $\gamma\in(0,1)$. In the latter case we show that, for dimensions $\alpha\gamma<d<\alpha(1+\gamma)$, the rescaled occupation time fluctuations converge weakly to a centered Gaussian process whose covariance function is explicitly calculated, and we call it {\em weighted sub-fractional Brownian motion.} Moreover, in the case of lifetimes with finite mean, we show that for $\alpha<d<2\alpha$ the fluctuation limit turns out to be the same as in the case of exponentially distributed lifetimes studied by Bojdecki et al. [7,8,9]. We also investigate the maximal parameter range allowing existence of the weighted sub-fractional Brownian motion and provide some of its fundamental properties, such as path continuity, long-range dependence, self-similarity and the lack of Markov property.