When are time changes of Feller-Dynkin processes still Feller-Dynkin processes? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T00:22:18Z http://mathoverflow.net/feeds/question/86873 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86873/when-are-time-changes-of-feller-dynkin-processes-still-feller-dynkin-processes When are time changes of Feller-Dynkin processes still Feller-Dynkin processes? ShawnD 2012-01-28T01:11:45Z 2012-01-30T14:25:31Z <p>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 <a href="http://www.cambridge.org/gb/knowledge/isbn/item1166383/?site_locale=en_GB" rel="nofollow">Rogers and Williams</a> or see <a href="http://en.wikipedia.org/wiki/Feller_process" rel="nofollow">Wikipedia</a>).</p> <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/86873/when-are-time-changes-of-feller-dynkin-processes-still-feller-dynkin-processes/87033#87033 Answer by John Dawkins for When are time changes of Feller-Dynkin processes still Feller-Dynkin processes? John Dawkins 2012-01-30T14:25:31Z 2012-01-30T14:25:31Z <p>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.</p>