Generalized space-time fractional dynamics in networks and lattices

TL;DR

This paper develops a flexible generalized space-time fractional random walk model on networks and lattices, capturing non-Markovian dynamics with long memory and fat-tailed waiting times, and derives a corresponding macroscopic diffusion equation.

Contribution

It introduces a novel four-parameter generalized space-time fractional CTRW model that unifies and extends classical models, applicable to complex systems.

Findings

Derived a macroscopic space-time fractional diffusion equation.

Analyzed special cases including Laskin's fractional Poisson process.

Demonstrated model's flexibility to describe real-world complex systems.

Abstract

We analyze generalized space-time fractional motions on undirected networks and lattices. The continuous-time random walk (CTRW) approach of Montroll and Weiss is employed to subordinate a space fractional walk to a generalization of the time-fractional Poisson renewal process. This process introduces a non-Markovian walk with long-time memory effects and fat-tailed characteristics in the waiting time density. We analyze `generalized space-time fractional diffusion' in the infinite $\it d$-dimensional integer lattice $\it \mathbb{Z}^d$. We obtain in the diffusion limit a `macroscopic' space-time fractional diffusion equation. Classical CTRW models such as with Laskin's fractional Poisson process and standard Poisson process which occur as special cases are also analyzed. The developed generalized space-time fractional CTRW model contains a four-dimensional parameter space and offers therefore a great flexibility to describe real-world situations in complex systems.