# When are time changes of Feller-Dynkin processes still Feller-Dynkin processes?

A Markov process $X_t$ on $E$ is a Feller-Dynkin (or sometimes just Feller) process if its semigroup is a strongly continuous, sub-Markov semigroup $\{P_t:t\geq 0\}$ of linear operators on $C_0(E)$ (my definition comes from the text of Rogers and Williams or see Wikipedia).

I am wondering what is known about when a time change of a Feller-Dynkin process is still Feller-Dynkin. I believe that I have read that this is not true in general (and can be a tricky thing to determine). I would be curious to see an example where this fails and also to know if the Feller-Dynkin property is preserved if the time change is nice enough.

For what it's worth, the time changes I am interested in are actually $C^{\infty}$. I do not need the Feller property to be preserved, but thinking about it made me realize I do not have a very good feeling for these sort of things.

• By the so-called Dambis, Dubins-Schwarz theorem every continous local martingale is a time-changed Wiener process. Now it is enough to take your favourite martingale which is not a Markov process. – Piotr Miłoś Jan 29 '12 at 10:57
• Piotr Miłoś: as you gave the answer in your comment, I would recommend that you add an answer to this question so that Shawn can accept it. Best Regards – The Bridge Jan 30 '12 at 13:15
• Yes, this sort of idea seems to be a useful source of counterexamples. – ShawnD Feb 14 '12 at 1:16

If you time change $X_t$ using a "clock" of the form $B_t = \int_0^t b(X_s) ds$, with $b$ a bounded positive continuous function on the state space of $X_t$ that is also bounded below away from $0$, then the time-changed process will be a Feller process. This follows directly from the Hille-Yosida theorem.