Timeline for Local time of Brownian motion + Lipschitz continuous function
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Feb 7, 2015 at 13:02 | history | edited | ofer zeitouni | CC BY-SA 3.0 |
Removed an incorrect part of the argument
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| Dec 31, 2014 at 8:46 | comment | added | Nina Holden | I posted a comment above with an already published paper implying the lemma. The paper use some similar ideas to the proof sketched above: The number of rectangles of side length $\sim N^{-1}$ containing both $-g$ and $B$, is bounded by $cN^{1/2+\epsilon}$ with high probability, and this is used to prove uniform continuity of $g\mapsto L^g(B)$ on a countable dense subset of $\mathrm{Lip}(M)$. | |
| Dec 28, 2014 at 1:13 | comment | added | Nina Holden | Thanks! I think the logarithmic correction is not necessary for the moments. Defining the Brownian motion $\tilde{B}$ by $\tilde{B}_t=\epsilon^{-1/2}B_{\epsilon t}$, and letting $L_t^x$ denote local time at $x$ at time $t$, we have $\epsilon^{-3/2}Z_\epsilon= \epsilon^{-1/2}\sup_x\int_0^1 1_{|\tilde{B}_s-x|<2\epsilon^{1/2}}\,ds \leq 4\sup_x L_1^x(\tilde{B})$. The $p$th moment of the right-hand side is bounded by $C_p$, see e.g. "(Semi-) martingale inequalities and local times" by Barlow and Yor | |
| Dec 27, 2014 at 8:42 | history | edited | ofer zeitouni | CC BY-SA 3.0 |
Added an interpolation that boosts from $\epsilon^{1/2}$ to $\epsilon$ up to log factors.
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| Dec 25, 2014 at 20:33 | history | edited | ofer zeitouni | CC BY-SA 3.0 |
added 233 characters in body
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| Dec 25, 2014 at 17:10 | comment | added | ofer zeitouni | I did not check step 1 carefully, and you are right, it is flawed. I think that it can be rescued (working with longer intervals), I'll check and will let you know. | |
| Dec 25, 2014 at 15:38 | comment | added | Nina Holden | Thanks Ofer. But why do we have $EM_\epsilon\leq C_1$? Defining the Brownian motion $\widehat{B}$ by $B_t=\epsilon \widehat{B}_{t\epsilon^{−2}}$ and change of variables $s=t\epsilon^{-2}$, we get $M_\epsilon=\sup_x \int_0^{ϵ^{-1}} 1_{|\widehat{B}_s−x|<2}ds$, which diverges when $\epsilon\rightarrow 0$. | |
| Dec 24, 2014 at 16:14 | history | answered | ofer zeitouni | CC BY-SA 3.0 |