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

Abstract : Let $(\mathcal{T},d)$ 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.
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https://hal.archives-ouvertes.fr/hal-01026487
Contributor : Xan Duhalde <>
Submitted on : Monday, July 21, 2014 - 4:42:42 PM
Last modification on : Friday, March 27, 2020 - 3:43:17 AM
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• HAL Id : hal-01026487, version 1
• ARXIV : 1407.5563

### Citation

Xan Duhalde. Uniform Hausdorff measure of the level sets of the Brownian tree.. 2014. ⟨hal-01026487⟩

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