Anomalous Diffusion Index for Lévy Motions

Abstract

In modelling complex systems as real diffusion processes it is common to analyse its diffusive regime through the study of approximating sequences of random walks. For the partial sums $S_n=\xi_1 + \xi_2 + \ldots + \xi_n$ one considers the approximating sequence of processes $X^{(n)}(t)= a_n (S_{[k_nt]}-b_n)$. Then, under sufficient smoothness requirements we have the convergence to the desired diffusion, $X^{(n)}(t) \to X(t)$. A key assumption usually presumed is the finiteness of the second moment, and, hence the validity of the Central Limit Theorem. Under anomalous diffusive regime the asymptotic behavior of S _n may well be non-Gaussian and $n^{-1}E(S^2_n) \to\infty$. Such random walks have been referred by physicists as Lévy motions or Lévy flights. In this work, we introduce an alternative notion to classify these regimes, the diffusion index $\gamma_X$. For some $\gamma^0_X$ properly chosen let $\gamma_X=\inf \{ \gamma:0 < \gamma\leq\gamma^0_X,\limsup_{t\to \infty}t^{-1}E|X(t)|^{1/\gamma} < \infty \}$. Relationship between $\gamma_X$, the infinitesimal diffusion coefficients and the diffusion constant will be explored. Illustrative examples as well as estimates, based on extreme order statistics, for $\gamma_X$ will also be presented.