Continuous time random walk and parametric subordination in fractional diffusion

TL;DR

This paper explores the connection between continuous-time random walks and fractional diffusion equations, introducing a parametric subordination method to generate sample paths of non-Markovian processes with fractional dynamics.

Contribution

It presents a novel approach to derive fractional diffusion processes from CTRW using parametric subordination, highlighting the subordinated structure and non-Markovian behavior.

Findings

Derived space-time fractional diffusion equations from CTRW

Introduced parametric subordination for process generation

Displayed sample paths for specific fractional cases

Abstract

The well-scaled transition to the diffusion limit in the framework of the theory of continuous-time random walk (CTRW)is presented starting from its representation as an infinite series that points out the subordinated character of the CTRW itself. We treat the CTRW as a combination of a random walk on the axis of physical time with a random walk in space, both walks happening in discrete operational time. In the continuum limit we obtain a generally non-Markovian diffusion process governed by a space-time fractional diffusion equation. The essential assumption is that the probabilities for waiting times and jump-widths behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. By what we call parametric subordination, applied to a combination of a Markov process with a positively oriented L\'evy process, we generate and display sample paths for some special cases.