  We consider a general class of integro-differential evolution equations which
includes the governing equation of the generalized grey Brownian motion and the
time- and space-fractional heat equation. We present a general relation between
the parameters of the equation and the distribution of the underlying
stochastic processes, as well as discuss different classes of processes
providing stochastic solutions of these equations. For a subclass of evolution
equations, containing Saigo-Maeda generalized time-fractional operators, we
determine the parameters of the corresponding processes explicitly. Moreover,
we explain how self-similar stochastic solutions with stationary increments can
be obtained via linear fractional L\'evy motion for suitable
pseudo-differential operators in space.
