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Levy flights and fractional Brownian motion (fBm) have become exemplars ofthe heavy tailed jumps and long-ranged memory seen in space physics andelsewhere. Natural time series frequently combine both effects, and LinearFractional Stable Motion (LFSM) is a model process of this type, combiningalpha-stable jumps with a memory kernel. In contrast complex physicalspatiotemporal diffusion processes where both the above effects compete-dubbed"ambivalent" by Brockmann et al (2006}-have for many years been modelled usingthe 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. Wederive the analogous kinetic equation for LFSM and show that it has a diffusioncoefficient with a power law in time rather than having a fractional timederivative like the CTRW. We develop earlier comments by Lutz (2001) on how fBmdiffers from its fractional time process counterpart. We go on to argue morephysically why LFSM and the FFCTRW might indeed be expected to differ, anddiscuss some preliminary results on the scaling of burst "sizes" and"durations" in LFSM time series, with applications to modelling existingobservations in space physics.

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