After answering [this question] (Expectation of a stopping time uniquely determined by a function) I was looking for the literature on the mean hitting/exit time for a discrete-time Markov process.
In Meyn and Tweedie, Durett and some other books I haven't found much information. Google also didn't help much for the discrete-time setting.
I guess that for the uncountable state space, mean hitting time equation shares the same properties as for the countable state space. However, I would prefer to read a book chapter on this topic.
Dynkin briefly discussed it, though for the continuous time and too briefly.