  We extend results on time-rescaled occupation time fluctuation limits of the
$(d,\alpha, \beta)$-branching particle system $(0<\alpha \leq 2, 0<\beta \leq
1)$ with Poisson initial condition. The earlier results in the homogeneous case
(i.e., with Lebesgue initial intensity measure) were obtained for dimensions
$d>\alpha / \beta$ only, since the particle system becomes locally extinct if
$d\le \alpha / \beta$. In this paper we show that by introducing high density
of the initial Poisson configuration, limits are obtained for all dimensions,
and they coincide with the previous ones if $d>\alpha/\beta$. We also give
high-density limits for the systems with finite intensity measures (without
high density no limits exist in this case due to extinction); the results are
different and harder to obtain due to the non-invariance of the measure for the
particle motion. In both cases, i.e., Lebesgue and finite intensity measures,
for low dimensions ($d<\alpha(1+\beta)/\beta$ and
$d<\alpha(2+\beta)/(1+\beta)$, respectively) the limits are determined by
non-L\'evy self-similar stable processes. For the corresponding high dimensions
the limits are qualitatively different: ${\cal S}'(R^d)$-valued L\'evy
processes in the Lebesgue case, stable processes constant in time on
$(0,\infty)$ in the finite measure case. For high dimensions, the laws of all
limit processes are expressed in terms of Riesz potentials. If $\beta=1$, the
limits are Gaussian. Limits are also given for particle systems without
branching, which yields in particular weighted fractional Brownian motions in
low dimensions. The results are obtained in the setup of weak convergence of
S'(R^d)$-valued processes.
