Superdiffusion and stable laws

The superdiffusion equation with a fractional Laplacian Delta(alpha/2) in N-dimensional space describes the asymptotic (t --> infinity) behavior of a generalized Poisson process with the range (discontinuity) distribution density similar to\x\(-alpha-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 alpha >1 merely amounts to a reduction in the diffusion coefficient with the form of the distribution remaining stable. For alpha < 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. (C) 1999 American Institute of Physics. [S1063-7761(99)01804-1].