Fractional calculus and stable probability distributions

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. Keywords -- Fractional calculus, diffusion equation, stable distributions, random-walk. 1. Introduction The purpose of this note is to outline the role of fractional calculus in generating stable probability distributions through generalized diffusion equations of fractional order. For the standard diffusion equation it is well known that the fundamental solution of the Cauchy problem provides the spatial probability density function (pdf) for the Gaussian or normal distribution, whose variance...

Перевод пока недоступен