Important developments in the study of stochastic processes with jumps have
been archived in recent yearsthrough the analytical approach. In thisapproach,
the (parabolic) Green function playsa key role. Asa product of our previous
book [44], here we collect several useful results derived from the construction of
the Green function with its suitable estimates, given in terms of the so-called
Green spaces. Special attention is paid to integro-differential problems with
oblique boundary conditions. One of the most important applications is then
to show the existence and uniqueness of the invariant measure by means of the
Green function. This invariant measure allows us to study the ergodic stopping
time and control problems.