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The $(d,\alpha,\beta,\gamma)$-branching particle system consists of particlesmoving in $R^d$ 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 measurewith intensity measure $\mu_\gamma(dx)=dx/(1+|x|^\gamma), \gamma\geq 0$. Bymeans of time rescaling $T$ and Poisson intensity measure $H_T\mu_\gamma$,occupation time fluctuation limits for the system as $T\to\infty$ have beenobtained in two special cases: Lebesgue measure ($\gamma=0$, the homogeneouscase), and finite measures $(\gamma>d)$. In some cases $H_T\equiv 1$ and inothers $H_T\to\infty$ as $T\to\infty$ (high density systems). The limitprocesses are quite different for Lebesgue and for finite measures. Thereforethe question arises of what kinds of limits can be obtained for Poissonintensity measures that are intermediate between Lebesgue measure and finitemeasures. In this paper the measures $\mu_\gamma, \gamma\in (0,d]$, are usedfor investigating this question. Occupation time fluctuation limits areobtained which interpolate in some way between the two previous extreme cases.The limit processes depend on different arrangements of the parameters$d,\alpha,\beta,\gamma$. Related results for the corresponding$(d,\alpha,\beta,\gamma)$-superprocess are also given.

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