Fractional diffusion equations and processes with randomly varying time

Abstract

In this paper the solutions u_ν=u_ν(x, t) to fractional diffusion equations of order 0<ν≤2 are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order ν=1/2ⁿ, 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 ν=2/3ⁿ, 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_ν coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions u_ν and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.