  We characterize the value function and the optimal stopping time for a large
class of optimal stopping problems where the underlying process to be stopped
is a fairly general Markov process. The main result is inspired by recent
findings for L\'evy processes obtained essentially via the Wiener-Hopf
factorization. The main ingredient in our approach is the representation of the
$\beta$-excessive functions as expected suprema. A variety of examples is
given.
