Random walk models approximating symmetric space-fractional diffusion processes

TL;DR

This paper introduces three discrete random walk models that approximate symmetric space-fractional diffusion processes, demonstrating their convergence to stable distributions and analyzing a specific model by Chechkin and Gonchar.

Contribution

It presents new discrete models for symmetric space-fractional diffusion and extends the inversion theory of Riesz potential operators to the critical case $lpha=1$.

Findings

Models converge to stable distributions as steps vanish

Analysis of Chechkin and Gonchar's model in detail

Extension of Riesz potential inversion to $lpha=1$

Abstract

For the symmetric case of space-fractional diffusion processes (whose basic analytic theory has been developed in 1952 by Feller via inversion of Riesz potential operators) we present three random walk models discrete in space and time. We show that for properly scaled transition to vanishing space and time steps these models converge in distribution to the corresponding time-parameterized stable probability distribution. Finally, we analyze in detail a model, discrete in time but continuous in space, recently proposed by Chechkin and Gonchar. Concerning the inversion of the Riesz potential operator $I_0^\alpha$ let us point out that its common hyper-singular integral representation fails for $\alpha = 1. In our Section 2 we have shown that the corresponding hyper-singular representation for the inverse operator $D_0^\alpha$ can be obtained also in the critical (often excluded) case $\alpha = 1$, by analytic continuation