Fractional Diffusion Processes: Probability Distributions and Continuous Time Random Walk

A physical-mathematical approach to anomalous diffusion may be based on fractional diffusion equations and related random walk models. The fundamental solutions of these equations can be interpreted as probability densities evolving in time of peculiar self-similar stochastic processes: an integral representation of these solutions is here presented. A more general approach to anomalous diffusion is known to be provided by the master equation for a continuous time random walk (CTRW). We show how this equation reduces to our fractional diffusion equation by a properly scaled passage to the limit of compressed waiting times and jump widths. Finally, we describe a method of simulation and display (via graphics) results of a few numerical case studies.