EXACT-SOLUTIONS FOR A CLASS OF FRACTAL TIME RANDOM-WALKS

Fractal time random walks with generalized Mittag-Leffler functions as waiting time densities are studied. This class of fractal time processes is characterized by a dynamical critical exponent 0 < omega less than or equal to 1, and is equivalently described by a fractional master equation with time derivative of noninteger order omega. Exact Greens functions corresponding to fractional diffusion are obtained using Mellin transform techniques. The Greens functions are expressible in terms of general H-functions. For omega < 1 they are singular at the origin and exhibit a stretched Gaussian form at infinity. Changing the order omega interpolates smoothly between ordinary diffusion omega = 1 and completely localized behavior in the omega --> 0 limit.