  As a class of L\'evy type Markov generators, nonlocal Waldenfels operators
appear naturally in the context of investigating stochastic dynamics under
L\'evy fluctuations and constructing Markov processes with boundary conditions
(in particular the construction with jumps). This work is devoted to prove the
weak and strong maximum principles for `parabolic' equations with nonlocal
Waldenfels operators. Applications in stochastic differential equations with
$\alpha$-stable L\'evy processes are presented to illustrate the maximum
principles.
