Revisiting the derivation of the fractional diffusion equation

TL;DR

This paper revisits the derivation of the fractional diffusion equation from continuous-time random walks, clarifying the diffusion limit and solving the associated Cauchy problem in various cases.

Contribution

It provides a clear derivation of the fractional diffusion equation from CTRWs using the Gnedenko-Kolmogorov limit theorem and discusses the proper diffusion limit.

Findings

Derived the fractional diffusion equation from CTRWs

Solved the Cauchy problem in multiple cases

Clarified the diffusion limit for CTRWs

Abstract

The fractional diffusion equation is derived from the master equation of continuous-time random walks (CTRWs) via a straightforward application of the Gnedenko-Kolmogorov limit theorem. The Cauchy problem for the fractional diffusion equation is solved in various important and general cases. The meaning of the proper diffusion limit for CTRWs is discussed.