Probability distributions generated by fractional diffusion equations

TL;DR

This paper explores how fractional diffusion equations generate probability distributions that are related to stable distributions, extending classical diffusion models for applications in finance and economics.

Contribution

It introduces the fundamental solutions of fractional diffusion equations as probability density functions linked to stable distributions, generalizing standard diffusion models.

Findings

Fundamental solutions are related to stable distributions.

Fractional derivatives generalize classical diffusion equations.

Applications in financial and economic modeling.

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 fractional order. The fundamental solutions of these equations provide probability density functions, evolving on time or variable in space, which are related to the class of stable distributions. This property is a noteworthy generalization of what happens for the standard diffusion equation and can be relevant in treating financial and economical problems where the stable probability distributions play a key role.