Fractional diffusion equations and processes with randomly varying time

Abstract

In this paper the solutions u v = u v(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 v = 1/2n, n > 1, we show that the solutions u1/2 n 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 v = 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 v coincides with the distribution of Brownian motion with random, time or of different processes with a Brownian time. The interplay between the solutions u v and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit. © Institute of Mathematical Statistics, 2009.