Timeline for Invariant and periodic measures of the random dynamical system on the circle generated by $d\theta_t=dW_t$
Current License: CC BY-SA 3.0
3 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 10, 2020 at 16:11 | comment | added | Julian Newman | @julian This is indeed true; "irreducibility" here can be taken to mean e.g. "for each $\lambda$-positive measure open set $U$, $P(x,U)>0$ for all $x$". In my proof, I used the much stronger fact that for every $\lambda$-positive measure Borel set $A$, $P(x,A)>0$ for all $x$. This does imply that $\lambda$ is the unique stationary measure, without requiring additional assumptions such as the strong Feller property. | |
| Feb 10, 2020 at 14:57 | comment | added | julian | To be utterly precise, irreducibility of the semigroup doesn't yet give you unique for the invariant measure. You need in addition strong Feller at some positive time. A famous counterexample would be the Ising model in dimension $d\geq 2$ above the critical temperature. | |
| Jul 16, 2015 at 0:37 | history | answered | Julian Newman | CC BY-SA 3.0 |