We show in this paper that the sample (time average) mean-squared displacement (MSD) of the fractional Lévy $\alpha$-stable motion behaves very differently from the corresponding ensemble average (second moment). While the ensemble average MSD diverges for $\alpha <2$, the sample MSD may exhibit either subdiffusion, normal diffusion, or superdiffusion. Thus, $H$-self-similar Lévy stable processes can model either a subdiffusive, diffusive or superdiffusive dynamics in the sense of sample MSD. We show that the character of the process is controlled by a sign of the memory parameter $d=H-1/\alpha$. We also introduce a sample $p$-variation dynamics test which allows to distinguish between two models of subdiffusive dynamics. Finally, we illustrate a subdiffusive behavior of the fractional Lévy stable motion on biological data describing the motion of individual fluorescently labeled mRNA molecules inside live E. coli cells, but it may concern many other fields of contemporary experimental physics.

• Received 23 April 2010

DOI:https://doi.org/10.1103/PhysRevE.82.021130