A compounded random walk for space-fractional diffusion on finite domains

TL;DR

This paper introduces a new compounded random walk model that accurately describes space-fractional diffusion on finite and infinite domains, with applications in modeling superdiffusive processes influenced by potential fields.

Contribution

It presents a physically well-defined compounded random walk that converges to a space-fractional Fokker-Planck equation on bounded domains, advancing modeling capabilities.

Findings

Excellent agreement between simulations and analytical solutions

Effective numerical approximation methods demonstrated

Model captures superdiffusive behavior on finite domains

Abstract

We formulate a compounded random walk that is physically well defined on both finite and infinite domains, and samples space-dependent forces throughout jumps. The governing evolution equation for the walk limits to a space-fractional Fokker-Planck equation valid on bounded domains, and recovers the well known superdiffusive space-fractional diffusion equation on infinite domains. We describe methods for numerical approximation and Monte Carlo simulations and demonstrate excellent correspondence with analytical solutions. This compounded random walk, and its associated fractional Fokker-Planck equation, provides a major advance for modeling space-fractional diffusion through potential fields and on finite domains.