  We consider generalized time-fractional evolution equations of the form
$$u(t)=u_0+\int_0^tk(t,s)Lu(s)ds$$ with a fairly general memory kernel $k$ and
an operator $L$ being the generator of a strongly continuous semigroup. In
particular, $L$ may be the generator $L_0$ of a Markov process $\xi$ on some
state space $Q$, or $L:=L_0+b\nabla+V$ for a suitable potential $V$ and drift
$b$, or $L$ generating subordinate semigroups or Schr\"{o}dinger type groups.
This class of evolution equations includes in particular time- and space-
fractional heat and Schr\"odinger type equations. We show that a subordination
principle holds for such evolution equations and obtain Feynman-Kac formulae
for solutions of these equations with the use of different stochastic
processes, such as subordinate Markov processes and randomly scaled Gaussian
processes. In particular, we obtain some Feynman-Kac formulae with generalized
grey Brownian motion and other related self-similar processes with stationary
increments.
