Dynamical approach to Levy processes

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 Phi(y)(t). Of special interest is that the distribution of the times of sojourn in the two states of the fluctuating process is proportional to d(2) Phi(y)(t)/dt(2). Furthermore, in the special case where Phi(y)(t) has an inverse power law, with the index beta ranging from 0 to 1, thus making it nonintegrable, we show analytically that the statistics of the diffusing variable approximate in the long-time limit the alpha-stable Levy distributions. The departure of the diffusion process of dynamical origin from the ideal condition of the Levy statistics is established by means of a simple analytical expression. We note, first of all, that the characteristic function of a genuine Levy 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 Phi(y)(t). Since the characteristic function can be given a spectroscopic significance, we also discuss the relevance of our results within this context.