Equivalent continuous and discrete realizations of Levy flights: A model of one-dimensional motion of an inertial particle

The paper is devoted to the relationship between the continuous Markovian description of Levy flights developed previously (see, e.g., I.A. Lubashevsky, Truncated Levy flights and generalized Cauchy processes, Eur. Phys. J. B 82 (2011) 189-195 and references therein) and their equivalent representation in terms of discrete steps of a wandering particle, a certain generalization of continuous time random walks. To simplify understanding the key points of the technique to be created, our consideration is confined to the one-dimensional model for continuous random motion of a particle with inertia. Its dynamics governed by stochastic self-acceleration is described as motion on the phase plane {x, v} comprising the position x and velocity v = dx/dt of the given particle. A notion of random walks inside a certain neighborhood L of the line v = 0 (the x-axis) and outside it is developed. It enables us to represent a continuous trajectory of particle motion on the plane {x, v} as a collection of the corresponding discrete steps. Each of these steps matches one complete fragment of the velocity fluctuations originating and terminating at the boundary of L. As demonstrated, the characteristic length of particle spatial displacement is mainly determined by velocity fluctuations with large amplitude, which endows the derived random walks along the x-axis with the characteristic properties of Levy flights. Using the developed classification of random trajectories a certain parameter-free core stochastic process is constructed. Its peculiarity is that all the characteristics of Levy flights similar to the exponent of the Levy scaling law are no more than the parameters of the corresponding transformation from the particle velocity v to the related variable of the core process. In this way the previously found validity of the continuous Markovian model for all the regimes of Levy flights is explained. Based on the obtained results an efficient single-peak approximation is constructed. In particular, it enables us to calculate the basic characteristics of Levy flights using the probabilistic properties of extreme velocity fluctuations and the shape of the most probable trajectory of particle motion within such extreme fluctuations. (C) 2013 Elsevier B.V. All rights reserved.