FRACTIONAL DIFFUSION EQUATION FOR FRACTAL-TIME CONTINUOUS-TIME RANDOM-WALKS

It is pointed out that the spherical symmetric diffusion equation of fractional order for the time, recently proposed by E.Roman and Giona [J.Phys.A 25(1992)2107-2117]: partial derivative(t)(1/dw)P(r,t) = -A [partial derivative/partial derivative P(r,t) + d-1/2r P(r,t)] corresponds, for integer spatial dimensions d = 1 and d = 3, to the long time description of a the fractal-time continuous-time random-walk, in which the interjump density has the asymptotic form psi(t) similar to t(-1-gamma), with 0 < gamma = 2/d(w) < 1.