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 ${\mathrm{\ensuremath{\Phi}}}_{\mathit{y}}$(t). Of special interest is that the distribution of the times of sojourn in the two states of the fluctuating process is proportional to ${\mathit{d}}^{2}$${\mathrm{\ensuremath{\Phi}}}_{\mathit{y}}$(t)/${\mathit{dt}}^{2}$. Furthermore, in the special case where ${\mathrm{\ensuremath{\Phi}}}_{\mathit{y}}$(t) has an inverse power law, with the index \ensuremath{\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 \ensuremath{\alpha}-stable L\'evy distributions. The departure of the diffusion process of dynamical origin from the ideal condition of the L\'evy statistics is established by means of a simple analytical expression. We note, first of all, that the characteristic function of a genuine L\'evy 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 ${\mathrm{\ensuremath{\Phi}}}_{\mathit{y}}$(t). Since the characteristic function can be given a spectroscopic significance, we also discuss the relevance of our results within this context. \textcopyright{} 1996 The American Physical Society.