Occupation times of branching systems with initial inhomogeneous Poisson states and related superprocesses

Abstract

The $(d,\alpha,\beta,\gamma)$-branching particle system consists of particles moving in $\mathbb{R}^d$ according to a symmetric $\alpha$-stable L\'evy process $(0<\alpha\leq 2)$, splitting with a critical $(1+\beta)$-branching law $(0<\beta\leq 1)$, and starting from an inhomogeneous Poisson random measure with intensity measure $\mu_\gamma(dx)=dx/(1+|x|^\gamma), \gamma\geq 0$. By means of time rescaling $T$ and Poisson intensity measure $H_T\mu_\gamma$, occupation time fluctuation limits for the system as $T\to\infty$ have been obtained in two special cases: Lebesgue measure ($\gamma=0$, the homogeneous case), and finite measures $(\gamma > d)$. In some cases $H_T\equiv 1$ and in others $H_T\to\infty$ as $T\to\infty$ (high density systems). The limit processes are quite different for Lebesgue and for finite measures. Therefore the question arises of what kinds of limits can be obtained for Poisson intensity measures that are intermediate between Lebesgue measure and finite measures. In this paper the measures $\mu_\gamma, \gamma\in (0,d]$, are used for investigating this question. Occupation time fluctuation limits are obtained which interpolate in some way between the two previous extreme cases. The limit processes depend on different arrangements of the parameters $d,\alpha,\beta,\gamma$. There are two thresholds for the dimension $d$. The first one, $d=\alpha/\beta+\gamma$, determines the need for high density or not in order to obtain non-trivial limits, and its relation with a.s. local extinction of the system is discussed. The second one, $d=[\alpha(2+\beta)-\gamma\vee \alpha]/\beta$\ (if $\gamma < d$), interpolates between the two extreme cases, and it is a critical dimension which separates different qualitative behaviors of the limit processes, in particular long-range dependence in ``low'' dimensions, and independent increments in ``high'' dimensions. In low dimensions the temporal part of the limit process is a new self-similar stable process which has two different long-range dependence regimes depending on relationships among the parameters. Related results for the corresponding $(d,\alpha,\beta,\gamma)$-superprocess are also given.