Mirko D’Ovidio

Fractional and anomalous diffusions have a long history. The terms fractional and anomalous have been considered with different meaning and in different contexts. By fractional diffusion we mean a diffusion in a medium with fractional dimension (fractals, for instance) whereas, by anomalous diffusions, according to the most significant literature, we refer to a motion whose mean squared displacement is proportional to a power of time. In this context we usually have the characterization given in terms of subdiffusion/superdiffusion or normal diffusion. Our aim is to pay exclusive attention to the probabilistic models for anomalous dynamics (not necessarily anomalous diffusions). The involved processes are guided by fractional (in time/space) equations and are obtained through random time changes.

We introduce a precise definition of delayed and rushed processes and provide some examples which are, in some cases, counter-intuitive.

We consider time changes given by subordinators and their inverse processes. Our analysis shows that, quite surprisingly, inverse processes are not necessarily leading to delayed processes.