Continuous time Markov processes on general state spaces

Question

Looking around I have found lots of material on continuous time Markov processes on finite or countable state spaces, say $\{0,1,\ldots,J\}$ for some $J\in\mathbb{N}$ or just $\mathbb{N}$. Similarly I have earlier worked with (discrete time) Markov chains on general state spaces, following the modern classic by Meyn & Tweedie.

My question concerns monographs on continuous time Markov processes on general state spaces, say some subset of $\mathbb{R}^k$, $k\in\mathbb{N}$. Are there any good references - preferably but not necessarily suited for an ambitious master student - on this topic?

Answer 1

I just finished a course on stochastic calculus, so I still lack experience on continuous time markov processes, but I think I can give you some references I have looked at:

-Chapter 10 of Introduction to Stochastic Integration by Hui-Hsiung Kuo explains the Markov property (and gives examples in terms of the Brownian motion), and diffusion processes. If you have a background in stochastic calculus (this book or Oksendal's are good introductory books) I would start here.

-A more detailed exposition (but too advanced for me at the moment of writing) is the two volume book of Rogers and Williams, Diffussions, Markov Processes and Martingales.

Answer 2

Another classic in probability theory is William Feller's An Introduction to Probability Theory and its Applications, volume I and II, which has a decent coverage of the topic, perhaps it is worth looking up (more specifically chapter 10, vol. II), even though it is a bit old.