From Power Laws to Fractional Diffusion: the Direct Way

TL;DR

This paper derives fractional diffusion equations from continuous time random walks with power-law waiting times and jumps, providing a link between random walk models and fractional PDEs with applications to simulations.

Contribution

It introduces a unified approach to obtain fractional diffusion processes from generalized random walks with fat-tailed distributions, including explicit solutions and simulation methods.

Findings

Derivation of fractional PDEs from random walk models.

Explicit integral representations of solutions.

Simulation techniques for fractional diffusion processes.

Abstract

Starting from the model of continuous time random walk, we focus our interest on random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for the waiting times, between 0 and 2 for the jumps. By stating the relevant lemmata (of Tauber type) for the distribution functions we need not distinguish between continuous and discrete space and time. We will see that, by a well-scaled passage to the diffusion limit, generalized diffusion processes, fractional in time as well as in space, are obtained. The corresponding equation of evolution is a linear partial pseudo-differential equation with fractional derivatives in time and in space, the orders being equal to the above exponents. Such processes are well approximated and visualized by simulation via various types of random walks. For their explicit solutions there are available integral representations that allow to investigate their detailed structure.