Truncated Lévy flights and generalized Cauchy processes

Abstract

A continuous Markovian model for truncated Lévy flights is proposed. It generalizes the approach developed previously by Lubashevsky et al. [Phys. Rev. E 79, 011110 (2009); Phys. Rev. E 80, 031148 (2009), Eur. Phys. J. B 78, 207 (2010)] and allows for nonlinear friction in wandering particle motion as well as saturation of the noise intensity depending on the particle velocity. Both the effects have own reason to be considered and, as shown in the paper, individually give rise to a cutoff in the generated random walks meeting the Lévy type statistics on intermediate scales. The nonlinear Langevin equation governing the particle motion was solved numerically using an order 1.5 strong stochastic Runge-Kutta method. The obtained numerical data were employed to analyze the statistics of the particle displacement during a given time interval, namely, to calculate the geometric mean of this random variable and to construct its distribution function. It is demonstrated that the time dependence of the geometric mean comprises three fragments following one another as the time scale increases that can be categorized as the ballistic regime, the Lévy type regime (superballistic, quasiballistic, or superdiffusive one), and the standard motion of Brownian particles. For the intermediate Lévy type part the distribution of the particle displacement is found to be of the generalized Cauchy form with cutoff. Besides, the properties of the random walks at hand are shown to be determined mainly by a certain ratio of the friction coefficient and the noise intensity rather than their characteristics individually.