Timeline for Null sets visited infinitely often by trajectories of the shift dynamical system
Current License: CC BY-SA 3.0
9 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Feb 22, 2014 at 19:03 | comment | added | Julian Newman | @BenoîtKloeckner: I agree that this would have been the obvious interpretation of "the event that you're at one particular point of the circle", if I had defined $\theta^t$ to be the "horizontal shift" on $C(\mathbb{R},G)$; however, I defined $\theta^t$ to be the "diagonal shift" on $C_0(\mathbb{R},G)$. Nonetheless, after reading Anthony's post, I did wonder whether for some fixed $\tau > 0$ and $x \in \mathbb{R}$ (or $x \in \mathbb{S}^1$) the set $A:=\{\omega \in \Omega | \omega(\tau)=x\}$ would work. For some reason I thought that it wouldn't - but I now realise that it probably does. | |
| Feb 22, 2014 at 16:33 | comment | added | Benoît Kloeckner | @JulianNewman: your example is of the same taste as Anthony Quas' one. "The event that you're at one particular point of the circle" means you fix $x\in\mathbb{S}^1$, (the "particular point") and then consider $A=\{\omega\in\Omega | \omega(0)=x\}$. It is very useful in probability to learn how to be able to translate between short sentences and full statements of this kind. | |
| Feb 22, 2014 at 13:52 | comment | added | Julian Newman | However, I think I have just managed to answer the main part of my question: Let $\mathbb{P}$ be the Wiener measure, and let $A$ be the set of all $\omega \in \Omega$ for which there exists $\varepsilon>0$ such that $\omega(t) \neq 0$ for all $t \in (0,\varepsilon)$. Then $A$ is a $\mathbb{P}$-null set, and $\Omega_A$ is a $\mathbb{P}$-full set. | |
| Feb 22, 2014 at 13:48 | comment | added | Julian Newman | @AnthonyQuas I don't see how "the event that you're at one particular point of the circle" is a well-defined subset of $\Omega$. | |
| S Feb 22, 2014 at 2:33 | history | suggested | gaoxinge | CC BY-SA 3.0 |
more clear
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| Feb 22, 2014 at 2:14 | review | Suggested edits | |||
| S Feb 22, 2014 at 2:33 | |||||
| Feb 21, 2014 at 17:33 | comment | added | Anthony Quas | What about Brownian motion on the circle; $A$ is the event that you're at one particular point of the circle? | |
| Feb 21, 2014 at 16:20 | history | asked | Julian Newman | CC BY-SA 3.0 |