1203.5860 (Ihor Lubashevsky)

Continuous and discrete realization of Levy flights. One-dimensional
process    [PDF]

Ihor Lubashevsky

The present paper is focused on constructing a relationship between continuous Markovian models for one-dimensional Levy flights as random motion of a wandering particle with stochastic self-acceleration and their discrete representation that may be treated as a generalized version of continuous time random walks (CTRW). For this purpose a notion of random motion inside a certain neighborhood of the particle velocity axis and outside it is developed. In this way a continuous particle trajectory is reduced to a collection of discrete steps of particle spatial displacement determined mainly by particle motion within individual peaks forming the time pattern of the velocity fluctuations. The obtained discrete random walks, indeed, may be treated as some generalization of CTRW because the individual duration of their steps and the corresponding particle displacement are random variables correlated in part with each other. The main difference between the standard approach and the constructed one is due to no assumption similar to uniform motion of a particle between the terminal points of one step is adopted. In addition, using the developed trajectory classification a certain parameter-free core stochastic process is constructed, so 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. In this way the validity of the continuous Markovian model for all the regimes of Levy flights is explained. Based on the obtained results an efficient "one-peak" approximation is constructed that enables one to find the basic characteristics of Levy flights using the extreme values of the velocity fluctuations and the shape of the most probable trajectories of particle motion.

View original: http://arxiv.org/abs/1203.5860