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