A kinetic equation for linear fractional stable motion with applications to space plasma physics

Abstract

Levy flights and fractional Brownian motion (fBm) have become exemplars of the heavy tailed jumps and long-ranged memory seen in space physics and elsewhere. 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-dubbed "ambivalent" by Brockmann et al (2006}-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 develop earlier comments by Lutz (2001) on how fBm differs from its fractional time process counterpart. We go on to argue more physically why LFSM and the FFCTRW might indeed be expected to differ, and 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.