Date: November 1996
Creator: Allegrini, Paolo; Grigolini, Paolo & West, Bruce J.
Description: This article discusses a dynamical approach to Lévy processes.Abstract: We derive the diffusion process generated by a correlated dichotomous fluctuating variable y starting from a Liouville-like equation by means of a projection procedure. This approach makes it possible to derive all statistical properties of the diffusion process from the correlation function of the dichotomous fluctuating variable Φy(t). Of special interest is that the distribution of the times of sojourn in the two states of the fluctuating process is proportional to d²Φy(t)/dt². Furthermore, in the special case where Φy(t) has an inverse power law, with the index β ranging from 0 to 1, thus making it nonintegrable, the authors show analytically that the statistics of the diffusing variable approximate in the long-time limit the α-stable Lévy distributions. The departure of the diffusion process of dynamical origin from the ideal condition of the Lévy statistics is established by means of a simple analytical expression. We note, first of all, that the characteristic function of a genuine Lévy process should be an exponential in time. We evaluate the correction to this exponential and show it to be expressed by a harmonic time oscillation modulated by the correlation function Φy(t). Since the characteristic function ...
Contributing Partner: UNT College of Arts and Sciences