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
6 events
| when toggle format | what | by | license | comment | |
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| Jul 21, 2015 at 16:13 | comment | added | Julian Newman | Yes, group extensions are what I had in mind. Specifically, we should have: Theorem. Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, let $\theta:\Omega\to\Omega$ be a $\mathbb{P}$-preserving measurable map, let $X$ be an abelian compact metrisable topological group with Haar measure $\lambda$, and let $R:\Omega\to X$ be a measurable function. Define the skew product $\Theta:(\omega,x)\mapsto(\theta(\omega),R(\omega)x)$. If $\mathbb{P}\times\lambda$ is $\Theta$-ergodic, then it is the only $\Theta$-invariant measure whose projection onto $\Omega$ coincides with $\mathbb{P}$. | |
| Jul 17, 2015 at 8:20 | comment | added | Anthony Quas | I think the natural level of generalization is a group extension - or possibly a quotient group extension of a dynamical system. A key property here is that the convolution of Haar measure with anything is Haar measure. | |
| Jul 16, 2015 at 13:45 | comment | added | Julian Newman | For future readers: the proof that $\mathbb{P}_W \times \lambda$ is ergodic can be found within my answer further below. [I should probably have said that the way I extend the ergodicity of $\mathbb{P}_W|_{\mathcal{F}_0^\infty} \times \lambda$ to the ergodicity of $\mathbb{P}_W \times \lambda$ is adapted from Theorem 1.7.2(i) of Ludwig Arnold's monograph Random Dynamical Systems (Springer, 1998).] | |
| Jul 16, 2015 at 11:59 | comment | added | Julian Newman | Wow, this is wonderful. It seems to me that this argument should completely generalise, and provide a much shorter proof of the theorem that I mentioned in my answer. | |
| Jul 16, 2015 at 11:57 | vote | accept | Julian Newman | ||
| Jul 16, 2015 at 9:30 | history | answered | Anthony Quas | CC BY-SA 3.0 |