  The foundations of the fractional diffusion equation are investigated based
on coupled and decoupled continuous time random walks (CTRW). For this aim we
find an exact solution of the decoupled CTRW, in terms of an infinite sum of
stable probability densities. This exact solution is then used to understand
the meaning and domain of validity of the fractional diffusion equation. An
interesting behavior is discussed for coupled memories (i.e., L\'evy walks).
The moments of the random walk exhibit strong anomalous diffusion, indicating
(in a naive way) the breakdown of simple scaling behavior and hence of the
fractional approximation. Still the Green function $P(x,t)$ is described well
by the fractional diffusion equation, in the long time limit.

Statistical Mechanics