  We introduce oscillatory analogues of fractional Brownian motion,
sub-fractional Brownian motion and other related long range dependent Gaussian
processes, we discuss their properties, and we show how they arise from
particle systems with or without branching and with different types of initial
conditions, where the individual particle motion is the so-called c-random walk
on a hierarchical group. The oscillations are caused by the discrete and
ultrametric structure of the hierarchical group, and they become slower as time
tends to infinity and faster as time approaches zero. We also give other
results to provide an overall picture of the behavior of this kind of systems,
emphasizing the new phenomena that are caused by the ultrametric structure as
compared with results for analogous models on Euclidean space.
