Kinetic equation of linear fractional stable motion and applications to modeling the scaling of intermittent bursts
Lévy flights and fractional Brownian motion have become exemplars of the heavy-tailed jumps and longranged
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 modeled using the fully fractional kinetic equation for the continuous-time random walk
CTRW, with power laws in the probability density functions 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 modeling existing observations
in space physics and elsewhere.