FRACTIONAL DIFFUSION AND LEVY STABLE PROCESSES

Anomalous diffusion in which the mean square distance between diffusin g quantities increases faster than linearly in ''time'' has been obser ved in all manner of physical and biological systems from macroscopic surface growth to DNA sequences. Herein we relate the cause of this no ndiffusive behavior to the statistical properties of an underlying pro cess using an exact statistical model. This model is a simple two-stat e 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 Levy alpha-stabl e processes in the superdiffusive domain (1/2<H<1).