  We deal with some extensions of the space-fractional diffusion equation,
which is satisfied by the density of a stable process (see Mainardi, Luchko,
Pagnini (2001)): the first equation considered here is obtained by adding an
exponential differential operator expressed in terms of the Riesz-Feller
derivative. We prove that this produces a random additional term in the
time-argument of the corresponding stable process, which is represented by the
so-called Poisson process with drift. Analogously, if we add, to the
space-fractional diffusion equation, a logarithmic differential operator
involving the Riesz-derivative, we obtain, as a solution, the transition
semigroup of a stable process subordinated by an independent gamma subordinator
with drift. Finally, we show that a non-linear extension of the
space-fractional diffusion equation is satisfied by the transition density of
the process obtained by time-changing the stable process with an independent
linear birth process with drift.
