Mean squared displacement in a generalized L\'evy walk model

Abstract

L\'evy walks represent a class of stochastic models (space-time coupled continuous time random walks) with applications ranging from the laser cooling to the description of animal motion. The initial model was intended for the description of turbulent dispersion as given by the Richardson's law. The existence of this Richardson's regime in the original model was recently challenged in the work by T. Albers and G. Radons, Phys. Rev. Lett. 120, 104501 (2018): the mean squared displacement (MSD) in this model diverges, i.e. does not exist, in the regime, where it presumably should reproduce the Richardson's law. In the supplemental material to this work the authors present (but do not investigate in detail) a generalized model interpolating between the original one and the Drude-like models known to show no divergences. In the present work we give a detailed investigation of the ensemble MSD in this generalized model, show that the behavior of the MSD in this model is the same (up to prefactiors) as in the original one in the domains where the MSD in the original model does exist, and investigate the conditions under which the MSD in the generalized model does exist or diverges. Both ordinary and aged situations are considered.