Subdiffusive discrete time random walks via Monte Carlo and subordination

TL;DR

This paper introduces a Monte Carlo simulation method for discrete time random walks with Sibuya power law waiting times, offering an alternative numerical approach to solve fractional subdiffusion equations with scalable computation time.

Contribution

It develops a Monte Carlo method for simulating discrete time random walks with Sibuya waiting times, providing a new numerical scheme for fractional subdiffusion equations.

Findings

Computation time scales as a power law with fractional exponent.

Provides explicit form of a subordinator for these random walks.

Offers an alternative approximate solution to fractional subdiffusion equations.

Abstract

A class of discrete time random walks has recently been introduced to provide a stochastic process based numerical scheme for solving fractional order partial differential equations, including the fractional subdiffusion equation. Here we develop a Monte Carlo method for simulating discrete time random walks with Sibuya power law waiting times, providing another approximate solution of the fractional subdiffusion equation. The computation time scales as a power law in the number of time steps with a fractional exponent simply related to the order of the fractional derivative. We also provide an explicit form of a subordinator for discrete time random walks with Sibuya power law waiting times. This subordinator transforms from an operational time, in the expected number of random walk steps, to the physical time, in the number of time steps.