  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.
