Renewal and memory origin of anomalous diffusion: a discussion of their joint action.

The adoption of the formalism of fractional calculus is an elegant way to simulate either subdiffusion or superdiffusion from within a renewal perspective where the occurrence of an event at a given time t does not have any memory of the events occurring at earlier times. We illustrate a physical model to assign infinite memory to renewal anomalous diffusion and we find (i) a condition where the simultaneous action of a renewal and a memory source of subdiffusion generates localization and (ii) a condition where they make subdiffusion weaker and superdiffusion emerge. We argue that our approach may provide important contributions to the current search to distinguish the renewal from the memory source of subdiffusion.

Memory effects in fractional Brownian motion with Hurst exponent H<1/3.

We study the regression to the origin of a walker driven by dynamically generated fractional Brownian motion (FBM) and we prove that when the FBM scaling, i.e., the Hurst exponent H<1/3 , the emerging inverse power law is characterized by a power index that is a compelling signature of the infinitely extended memory of the system. Strong memory effects leads to the relation H=θ/2 between the Hurst exponent and the persistent exponent θ , which is different from the widely used relation H=1-θ . The latter is valid for 1/3