Suppose one is interested in stochastic processes for the purpose getting a theoretical understanding of MCMC. They already have a decent understanding of probability theory (let's say at the level Billingsley) and want to develop enough tools to potentially prove theorems about the Markov Chains they construct at a research level.
What book or sequence of books would be reasonable for getting up to speed on the theory?
EDIT: This is not a question about applying MCMC. It is essentially asking for a mathematics textbook. Something less like Robert and Casella and more like Meyn and Tweedie (an answer I probably would have accepted when I asked this question).