FRACTIONAL DIFFUSION EQUATIONS AND PROCESSES WITH RANDOMLY VARYING TIME

In this paper the solutions u(nu) = u(nu) (x, t) to fractional diffusion equations of order 0 < v <= 2 are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order nu = 1/2(n), n >= 1, we show that the solutions u(1/2n) correspond to the distribution of the n-times iterated Brownian motion. For these processes the distributions of the maximum and of the sojourn time are explicitly given. The case of fractional equations of order nu = 2/3(n), n >= 1, is also investigated and related to Brownian motion and processes with densities expressed in terms of Airy functions. In the general case we show that u(nu) coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions u(nu) and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.