  We consider particle systems in locally compact Abelian groups with particles
moving according to a process with symmetric stationary independent increments
and undergoing one and two levels of critical branching. We obtain long time
fluctuation limits for the occupation time process of the one-and two-level
systems. We give complete results for the case of finite variance branching,
where the fluctuation limits are Gaussian random fields, and partial results
for an example of infinite variance branching, where the fluctuation limits are
stable random fields. The asymptotics of the occupation time fluctuations are
determined by the Green potential operator G of the individual particle motion
and its powers $G^2, G^3$, and by the growth as $t\to\infty$ of the operator
$G_t=\int^t_0T_sds$ and its powers, where $T_t$ is the semigroup of the motion.
The results are illustrated with two examples of motions: the symmetric
$\alpha$-stable L\'evy process in $\erre^d$ $(0<\alpha\leq2)$,and the so called
c-hierarchical random walk in the hierarchical group of order N (0<c<N). We
show that the two motions have analogous asymptotics of $G_t$ and its powers
that depend on an order parameter $\gamma$ for their transience /recurrence
behavior. This parameter is $\gamma=d/\alpha-1$ for the $\alpha$-stable motion,
and $\gamma=\log c/\log (N/c)$ for the c-hierarchical random walk. As a
consequence of these analogies, the asymptotics of the occupation time
fluctuations of the corresponding branching particle systems are also
analogous. In the case of the c-hierarchical random walk, however, the growth
of $G_t$ and its powers is modulated by oscillations on a logarithmic time
scale.
