# Continuous time Markov processes on general state spaces

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?

• Then, it sounds like you need to write this monograph. – Michael Oct 8 '16 at 2:24
• As a first cut, if you assume the transition rates $\nu(x)$ out of each state $x$ are bounded, so that there are values $\nu_{min}, \nu_{max}$ such that $0 < \nu_{min} \leq \nu(x) \leq \nu_{max} < \infty$ for all $x \in \mathcal{S}$, then the embedded chain is a discrete time chain (on a general state space $\mathcal{S}$) and it seems that recurrence properties of the embedded discrete time chain would carry over to the continuous time version. The avg number of jumps to a set of states, as measured in the embedded chain, cannot differ too much from the fraction of time being in those states. – Michael Oct 8 '16 at 3:18