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 den sity similar to\x\(-alpha-1). The solutions of this equation belong to a cl ass of spherically symmetric stable distributions. The main properties of t hese solutions are given together with their representations in the form of integrals and series and the results of numerical calculations. It is show n 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 veloc ity of free particle motion and the superdiffusion equation becomes physica lly meaningless. (C) 1999 American Institute of Physics. [S1063-7761(99)018 04-1].