Parametric subordination in fractional diffusion processes

TL;DR

This paper develops a method for simulating one-dimensional space-time fractional diffusion processes using parametric subordination, enabling accurate path generation through discretization based on the infinite divisibility of stable subordinators.

Contribution

It introduces a novel parametric subordination approach derived from fractional PDEs, allowing precise simulation of particle paths in fractional diffusion.

Findings

Path simulation accuracy improves with finer discretization.

The method leverages the infinite divisibility of stable subordinators.

It provides a constructive way to generate particle trajectories in fractional diffusion.

Abstract

We consider simulation of spatially one-dimensional space-time fractional diffusion. Whereas in an earlier paper of ours we have developed the basic theory of what we call parametric subordination via three-fold splitting applied to continuous time random walk with subsequent passage to the diffusion limit, here we go the opposite way. Via Fourier-Laplace manipulations of the relevant fractional partial differential equation of evolution we obtain the subordination integral formula that teaches us how a particle path can be constructed by first generating the operational time from the physical time and then generating in operational time the spatial path. By inverting the generation of operational time from physical time we arrive at the method of parametric subordination. Due to the infinite divisibility of the stable subordinator, we can simulate particle paths by discretization where the generated points of a path are precise snapshots of a true path. By refining the discretization more and more fine details of a path become visible.