Timeline for Is the nearest walk to Brownian motion uniform?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Sep 13, 2010 at 5:39 | comment | added | John Jiang | The problem seems that the space of PL paths is not Hausdorff under uniform metric. But the choice of $2^n$ paths makes cells well-defined and the boundaries associated with them are indeed measure 0. | |
| Sep 12, 2010 at 16:58 | comment | added | Bjørn Kjos-Hanssen | @Bill Thurston: Thanks! I think this settles it. I have posted a follow-up question (how different are the two distributions as $n\rightarrow\infty$) separately: mathoverflow.net/questions/38481/… | |
| Sep 12, 2010 at 15:54 | vote | accept | Bjørn Kjos-Hanssen | ||
| Sep 12, 2010 at 15:40 | comment | added | fedja | That would be true if the paths had no common pieces, which, in our case, they do more often than not. Just look at what happens if two paths have a common piece and their deviation from $W$ is largest inside that piece. | |
| Sep 12, 2010 at 15:13 | comment | added | Bill Thurston | Cell boundaries have meaure 0. For any two PL paths $\alpha, \beta$, the probability distribution for the the difference of sup distances of $W$ to the pair of paths is a nice, absolutely continuous measure, with computable piecewise-analytic density. (although the first derivative of the density is discontinuous at any value that is the $\pm$distance between parallel segments of $\alpha, \beta$). Therefore, the probability of the difference being 0 is 0. | |
| Sep 12, 2010 at 13:57 | comment | added | fedja | The words "Voronoi subdivision" make sense only if the cell boundaries have measure zero, which is very far from the truth here. On the other hand, I also think that each of continuum conjectures of this type (why not to look at all the functions that have the distance to $W$ not more than $a$ times the optimal with $a>1$ instead, say) has 0 probability to be true. I'll give it more thought when I have more time. | |
| Sep 12, 2010 at 13:27 | history | answered | Bill Thurston | CC BY-SA 2.5 |