  In-situ spacecraft observations recently suggested that the transport of
energetic particles accelerated at heliospheric shocks can be anomalous, i.e.
the mean square displacement can grow non-linearly in time. In particular, a
new analysis technique has permitted the study of particle transport properties
from energetic particle time profiles upstream of interplanetary shocks.
Indeed, the time/spatial power laws of the differential intensity upstream of
several shocks are indicative of superdiffusion. A complete determination of
the key parameters of superdiffusive transport comprises the power-law index,
the superdiffusion coefficient, the related transition scale at which the
energetic particle profiles turn to decay as power laws, and the energy
spectral index of the shock accelerated particles. Assuming large-scale spatial
homogeneity of the background plasma, the power-law behaviour can been derived
from both a (microscopic) propagator formalism and a (macroscopic) fractional
transport equation. We compare the two approaches and find a relation between
the diffusion coefficients used in the two formalisms. Based on the assumption
of superdiffusive transport, we quantitatively derive these parameters by
studying energetic particle profiles observed by the Ulysses and Voyager 2
spacecraft upstream of shocks in the heliosphere, for which a superdiffusive
particle transport has previously been observed. Further, we have jointly
studied the electron energy spectra, comparing the values of the spectral
indices observed with those predicted by the standard diffusive shock
acceleration theory and by a model based on superdiffusive transport. For a
number of interplanetary shocks and for the solar wind termination shock, for
the first time we obtain the anomalous diffusion constants and the scale at
which the probability of particle free paths changes to a power-law...
