Fractional calculus and stable probability distributions

Abstract

Fractional calculus allows one to generalize the linear (one-dimensional) diffusion equation by replacing either the first time-derivative or the second space-derivative by a derivative of a fractional order. The fundamental solutions of these generalized diffusion equations are shown to provide certain probability density functions, in space or time, which are related to the relevant class of stable distributions. For the space fractional diffusion, a random-walk model is also proposed.