  Recent in-situ and remote observations suggest that the transport regime
associated with shock accelerated particles may be anomalous {i.e., the Mean
Square Displacement (MSD) of such particles scales non-linearly with time}. We
use self-consistent, hybrid PIC plasma simulations to simulate a quasi-parallel
shock with parameters compatible with heliospheric shocks, and gain insights
about the particle transport in such a system. For suprathermal particles
interacting with the shock we compute the MSD separately in the upstream and
downstream regions. Tracking suprathermal particles for sufficiently long times
up and/or downstream of the shock poses problems in particle plasma
simulations, such as statistically poor particle ensembles and trajectory
fragments of variable length in time. Therefore, we introduce the use of
time-averaged mean square displacement (TAMSD), which is based on single
particle trajectories, as an additional technique to address the transport
regime for the upstream and downstream regions. MSD and TAMSD are in agreement
for the upstream energetic particle population, and both give a strong
indication of superdiffusive transport, consistent with interplanetary shock
observations. MSD and TAMSD are also in reasonable agreement downstream, where
indications of anomalous transport are also found. TAMSD shows evidence of
heterogeneity in the diffusion properties of the downstream particle
population, ranging from subdiffusive behaviour of particles trapped in the
strong magnetic field fluctuations generated at the shock, to superdiffusive
behaviour of particles transmitted and moving away from the shock.
