Fractional diffusion and Lévy stable processes

Abstract

Anomalous diffusion in which the mean square distance between diffusing quantities increases faster than linearly in ``time'' has been observed in all manner of physical and biological systems from macroscopic surface growth to DNA sequences. Herein we relate the cause of this nondiffusive behavior to the statistical properties of an underlying process using an exact statistical model. This model is a simple two-state process with long-time correlations and is shown to produce a random walk described by an exact fractional diffusion equation. Fractional diffusion equations describe anomalous transport and are shown to have exact solutions in terms of Fox functions, including L\'evy \ensuremath{\alpha}-stable processes in the superdiffusive domain (1/21).