Fractional diffusion equations and processes with randomly varying time

In this paper the solutions $u_ν=u_ν(x,t)$ to fractional diffusion equations of order $0<ν\leq 2$ are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order $ν=\frac{1}{2^n}$, $n\geq 1,$ we show that the solutions $u_{1/2^n}$ 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 $ν=\frac{2}{3^n}$, $n\geq 1,$ 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.

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