Superdiffusion and stable laws

Abstract

The superdiffusion equation with a fractional Laplacian Δ α /2 in N -dimensional space describes the asymptotic ( t →∞) behavior of a generalized Poisson process with the range (discontinuity) distribution density ∼| x |− α −1. The solutions of this equation belong to a class of spherically symmetric stable distributions. The main properties of these solutions are given together with their representations in the form of integrals and series and the results of numerical calculations. It is shown that allowance for the finite velocity of free particle motion for α >1 merely amounts to a reduction in the diffusion coefficient with the form of the distribution remaining stable. For α <1 the situation changes radically: the expansion velocity of the diffusion packet exceeds the velocity of free particle motion and the superdiffusion equation becomes physically meaningless.