In this work we show that protons can exhibit both superdiffusive and ballistic propagation, at variance with standard diffusion. We carry out an analysis of impulsive solar energetic particle (SEP) events, for which the observed time profile of energetic particle fluxes represent the propagator of the corresponding transport equation. We show that in the case of superdiffusive or ballistic transport the propagator in the time asymptotic regime has a power law form, and that a fit of the observed time profiles allows to determine the transport regime. Using data obtained from ACE and SoHO spacecraft, two proton and electron events, which exhibit both superdiffusive and ballistic transport, will be shown. The finding of these anomalous regimes implies that no finite mean free path can be defined.