Timeline for How to evaluate the wiener measure of sets?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Aug 3, 2014 at 9:27 | vote | accept | supersnail | ||
| Jul 31, 2014 at 16:57 | answer | added | Kostya_I | timeline score: 1 | |
| Jul 14, 2014 at 15:25 | comment | added | Nate Eldredge | @supersnail: Perhaps you should rephrase your question, then, to focus on the specific problem that interests you (in particular the title is misleading). But the necessary techniques will depend on what $G$ is. For example, if $G = \mathbb{R}^n - \{p\}$, you are asking about the recurrence and transience of Brownian motion, and that's already a nontrivial fact. | |
| Jul 14, 2014 at 8:25 | comment | added | supersnail | @Nate Eldredge: I want to know the measure of the set G (which I wrote down above), which is quite explicit. Don"t you think so? | |
| Jul 14, 2014 at 8:24 | comment | added | supersnail | @Kjos-Hanssen: Why do you mean the set can't be a Borel set? | |
| Jul 14, 2014 at 8:23 | comment | added | supersnail | @Martin Hairer: Thank you very much for your comment. I will consider the construction via Kolmogorov soon. But I'm not a probabilist. | |
| Jul 14, 2014 at 1:53 | comment | added | Nate Eldredge | I don't think you are really asking the right question here. Asking "In general, how do I compute the Wiener measure of sets" is tantamount to asking "In general, how do I prove theorems about Brownian motion." It is a very large theory and there are a great variety of techniques. Some very deep and difficult theorems (as well as open questions) can be stated as asking for the Wiener measure of a particular easily-described Borel set. So if there's a specific set whose measure you want to know, you should ask that by itself. | |
| Jul 13, 2014 at 13:05 | answer | added | Matthias Ludewig | timeline score: 2 | |
| Jul 13, 2014 at 12:53 | comment | added | Bjørn Kjos-Hanssen | There is something wrong with the definition, as $ w (t) $ cannot belong to a set of $ w $'s. | |
| Jul 13, 2014 at 12:35 | comment | added | Martin Hairer | The usual construction goes via Kolmogorov's extension theorem, followed by Kolmogorov's continuity test. This gives you a Borel measure on the space of continuous functions. The set $\{w\,:\, w(t) \in G\,\forall t\}$ is a perfectly nice Borel set there. In order for it to have non-zero measure, you will need $n \ge 3$ and $G$ unbounded. | |
| Jul 13, 2014 at 9:20 | history | asked | supersnail | CC BY-SA 3.0 |