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 $g(r)=r\log \log 1/r$, 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 $a\in (0,\infty )$, ${\ell }^a(\cdot )=\frac{1}{2}{\mathcal{H}}_g(\cdot \cap \mathcal{T}(a))$, where ${\mathcal{H}}_g$ stands for the $g$-Hausdorff measure.

31 pages

Cited by in corpus (1)

• Uniform multifractal structure of stable trees