Green's function of a Markov Chain, and maybe of a Feller Process?

Question

How are the Green's functions of a Markov chain related to the notion from PDE theory?

For instance, if the Markov chain (i.e. discrete state space) is continuous time, then the Green's function I'm thinking of is $G(x, y)=\int_0^\infty p_t(x, y)dt$ where $p_t$ is the transition kernel.

Is there a reasonable notion of the green's "function" when you have a Feller process instead of a Markov chain? (So now instead of transition functions, you have semigroups, and instead of Q matrices one has a generator.)

Answer 1

How are the Green's functions of a Markov chain related to the notion from PDE theory?

These are the same: the Green function of a Markov chain is the Green function "from PDE theory" associated to its generator.

Is there a reasonable notion of the (G)reen's "function" when you have a Feller process instead of a Markov chain?

Indeed there is, with no scare quotes. For an introduction, see Liggett's book Continuous Time Markov Processes: An Introduction, especially chapter 3.