Some recent advances in theory and simulation of fractional diffusion processes

TL;DR

This paper reviews recent theoretical and simulation advances in fractional diffusion processes, focusing on limits from random walks, universality of waiting times, and particle trajectory generation methods.

Contribution

It introduces new insights into the limit processes, universality laws, and a novel parametric subordination method for simulating fractional diffusion.

Findings

Limit from random walks to space-time fractional diffusion established

Universality of Mittag-Leffler waiting times demonstrated

A new method for generating particle trajectories introduced

Abstract

To offer a view into the rapidly developing theory of fractional diffusion processes we describe in some detail three topics of present interest: (i) the well-scaled passage to the limit from continuous time random walk under power law assumptions to space-time fractional diffusion, (ii) the asymptotic universality of the Mittag-Leffler waiting time law in time-fractional processes, (iii) our method of parametric subordination for generating particle trajectories.