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 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 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.

Figure 1

(Color online) Same as Fig. 3, dependence of duration exponent $\beta$ on $H$, but one trial only.

Figure 2

(Color online) Same as Fig. 4, dependence of size exponent $\gamma$ on $H$, again one trial only.

Figure 3

(Color online) Dependence of exponent $\beta$ for pdf $p\left(\tau \right)$ of a burst of duration $\tau$ on $H$ for simulated lfsm in the fBm, $\alpha =2$ limit. The average of seven trials was taken.

Figure 4

(Color online) Dependence of exponent $\gamma$ for the pdf $p\left(s\right)$ of a burst of size $s$ on $H$ for simulated lfsm in fBm limit. Again an average of seven trials was taken.

Figure 5

(Color online) Dependence of exponent $\psi$ on $H$ in fBm case. $\psi$ captures growth of burst size $s$ with duration $\tau$.

Figure 6

(Color online) Same as Fig. 3 (burst duration exponent $\beta$ vs $H$; seven trials), but for $\alpha =1.8$.

Figure 7

(Color online) Same as Fig. 4 (burst size exponent $\gamma$ vs $H$; seven trials), but for $\alpha =1.8$.

Figure 8

(Color online) Same as Fig. 3 (burst duration exponent $\beta$ vs $H$; seven trials), but for $\alpha =1.6$.

Figure 9

(Color online) Same as Fig. 4 (burst size exponent $\gamma$ vs $H$; seven trials), but for $\alpha =1.6$.

Figure 10

(Color online) Same as Fig. 3 (burst duration exponent $\beta$ vs $H$; seven trials), but for $\alpha =1$.

Figure 11

(Color online) Same as Fig. 4 (burst size exponent $\gamma$ vs $H$; seven trials), but for $\alpha =1$.