Timeline for Probability that planar Brownian motion doesn't "encircle" 0
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 2, 2015 at 0:14 | answer | added | Greg Lawler | timeline score: 10 | |
| Sep 18, 2015 at 17:17 | answer | added | Nate Eldredge | timeline score: 3 | |
| Sep 18, 2015 at 6:52 | answer | added | Iosif Pinelis | timeline score: 4 | |
| Sep 18, 2015 at 2:04 | comment | added | Will Sawin | Clearly $\alpha$ can be at most $1/2$, because $\mathbb P^x(E) \geq \sqrt{x}$. View the random walk as taking place in the complex plane, with $x$ a real number. Consider the harmonic function $\operatorname{Re}\sqrt{B_t}$ taking the principal branch with a cut at the negative real axis. This is a martingale until $B_t$ touches the negative real axis, takes a value of $0$ on that axis, at most $1$ on the unit circle, and is $\sqrt{x}$ at the start, so the probability that $B_t$ touches the unit circle without ever touching the negative real axis is at least $\sqrt{x}$. | |
| Sep 18, 2015 at 1:49 | comment | added | Will Sawin | Does the Brownian motion start at $x$? | |
| Sep 18, 2015 at 1:15 | comment | added | Joseph O'Rourke | Another related question: "Twisted random walks." | |
| Sep 18, 2015 at 0:23 | answer | added | ofer zeitouni | timeline score: 9 | |
| Sep 17, 2015 at 21:56 | comment | added | Anthony Quas | In some sense, the question is really asking for the probability that BM does encircle 0 (i.e. what is really needed is a lower bound for the probability that 0 gets encircled). | |
| Sep 17, 2015 at 21:14 | comment | added | Nate Eldredge | Related: mathoverflow.net/questions/202944/… | |
| Sep 17, 2015 at 21:13 | comment | added | Nate Eldredge | I changed the title to something a little more self-contained. Feel free to edit further. | |
| Sep 17, 2015 at 21:13 | history | edited | Nate Eldredge | CC BY-SA 3.0 |
better title?
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| Sep 17, 2015 at 21:03 | comment | added | Nate Eldredge | I guess a preliminary question would be, how to see that $E$ is measurable? | |
| Sep 17, 2015 at 19:11 | review | First posts | |||
| Sep 17, 2015 at 19:12 | |||||
| Sep 17, 2015 at 19:07 | history | asked | user71299 | CC BY-SA 3.0 |