Uniform Hausdorff measure of the level sets of the Brownian tree

TL;DR

This paper establishes a precise almost sure relationship between the local time measure on level sets of the Brownian tree and the corresponding Hausdorff measure with a specific gauge function, uniformly across all levels.

Contribution

It proves a uniform, almost sure equivalence between the local time measure and the Hausdorff measure on all level sets of the Brownian tree, specifying the exact multiplicative constant.

Findings

Almost sure uniform equivalence between local time and Hausdorff measure.

Identification of the exact multiplicative constant as 1/2.

Extension of previous fixed-level results to all levels simultaneously.

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.