  From the point of view of stochastic analysis the Caputo and
Riemann-Liouville derivatives of order $\al \in (0,2)$ can be viewed as
(regularized) generators of stable L\'evy motions interrupted on crossing a
boundary. This interpretation naturally suggests fully mixed, two-sided or even
multidimensional generalizations of these derivatives, as well as a
probabilistic approach to the analysis of the related equations. These
extensions are introduced and some well-posedness results are obtained that
generalize, simplify and unify lots of known facts. This probabilistic analysis
leads one to study a class of Markov processes that can be constructed from any
given Markov process in $\R^d$ by blocking (or interrupting) the jumps that
attempt to cross certain closed set of 'check-points'.
