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Uniform Hausdorff measure of the level sets of the Brownian tree.

By Xan Duhalde

Abstract

31 pagesLet $(\mathcal{T},d)$ be the random real tree with root $\rho$ coded by a Brownian excursion. So $(\mathcal{T},d)$ is (up to normalisation) Aldous CRT \cite{AldousI} (see Le Gall \cite{LG91}). The $a$-level set of $\mathcal{T}$ is the set $\mathcal{T}(a)$ of all points in $\mathcal{T}$ that are at distance $a$ from the root. We know from Duquesne and Le Gall \cite{DuLG06} that for any fixed $a\in (0, \infty)$, the measure $\ell^a$ that is induced on $\mathcal{T}(a)$ by the local time at $a$ of the Brownian excursion, is equal, up to a multiplicative constant, to the Hausdorff measure in $\mathcal{T}$ with gauge function $g(r)= r \log\log1/r$, restricted to $\mathcal{T}(a)$. As suggested by a result due to Perkins \cite{Per88,Per89} for super-Brownian motion, we prove in this paper a more precise statement that holds almost surely uniformly in $a$, and we specify the multiplicative constant. Namely, we prove that almost surely for any $a\in (0, \infty)$, $\ell^a(\cdot) = \frac{1}{2} \mathscr{H}_g (\, \cdot \, \cap \mathcal{T}(a))$, where $\mathscr{H}_g$ stands for the $g$-Hausdorff measure

Topics: local time, Brownian tree, Hausdorff measure, CRT, Brownian excursion, local time., 60G57, 60J80, 28A78, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR]
Publisher: HAL CCSD
Year: 2014
OAI identifier: oai:HAL:hal-01026487v1
Provided by: Hal-Diderot

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