  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.
