On the kinetic equation of linear fractional stable motion and applications to modeling the scaling of intermittent bursts

TL;DR

This paper derives a kinetic equation for linear fractional stable motion (lfsm), revealing a power-law diffusion coefficient, and explores its applications in modeling burst scaling in physical systems with heavy tails and memory effects.

Contribution

It introduces a new kinetic equation for lfsm, contrasting it with the CTRW model, and applies it to analyze burst scaling in physical time series.

Findings

Kinetic equation for lfsm with power-law diffusion coefficient

Preliminary results on burst size and duration scaling

Applications to space physics and complex diffusion processes

Abstract

Levy flights and fractional Brownian motion (fBm) have become exemplars of the heavy tailed jumps and long-ranged memory widely seen in physics. Natural time series frequently combine both effects, and linear fractional stable motion (lfsm) is a model process of this type, combining alpha-stable jumps with a memory kernel. In contrast complex physical spatiotemporal diffusion processes where both the above effects compete have for many years been modelled using the fully fractional (FF) kinetic equation for the continuous time random walk (CTRW), with power laws in the pdfs of both jump size and waiting time. We derive the analogous kinetic equation for lfsm and show that it has a diffusion coefficient with a power law in time rather than having a fractional time derivative like the CTRW. We discuss some preliminary results on the scaling of burst "sizes" and "durations" in lfsm time series, with applications to modelling existing observations in space physics and elsewhere.