EPJ manuscript No. (will be inserted by the editor) Truncated Lévy Random Walks and Generalized Cauchy Processes

Abstract. A continuous Markovian model for truncated Lévy random walks is proposed. It generalizes the approach developed previously by Lubashevsky et al. Phys. Rev. E 79, 011110 (2009); 80, 031148 (2009), Eur. Phys. J. B 78, 207 (2010) allowing for nonlinear friction in wondering particle motion and saturation of the noise intensity depending on the particle velocity. Both the effects have own reason to be considered and individually give rise to truncated Lévy random walks as shown in the paper. The nonlinear Langevin equation governing the particle motion was solved numerically using an order 1.5 strong stochastic Runge-Kutta method and the obtained numerical data were employed to calculate the geometric mean of the particle displacement during a certain time interval 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 then their characteristics individually. Key words. Lévy random walks – cutoff – generalized Cauchy processes – nonlinear stochastic differential equation – nonlinear friction – noise intensity saturation – truncated power-law distribution – geometric mean – intermediate asymptotics PACS. 05.40.Fb Random walks and Levy flights – 02.50.Ga Markov processes – 02.50.Ey Stochastic processes – 05.10.Gg Stochastic analysis methods