2014/07/21 by Xan Duhalde, Duhalde, Xan
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR
paper · pdf · doi:10.48550/arxiv.1407.5563
31 pages
arxiv created 2014/07/21 · openalex publication_date 2014/07/21 · arxiv updated 2014/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (T,d) be the random real tree with root ρ coded by a Brownian excursion. So (T,d) is (up to normalisation) Aldous CRT \citeAldousI (see Le Gall \citeLG91). The a-level set of T is the set T(a) of all points in T that are at distance a from the root. We know from Duquesne and Le Gall \citeDuLG06 that for any fixed a∈ (0, ∞), the measure ℓa that is induced on T(a) by the local time at a of the Brownian excursion, is equal, up to a multiplicative constant, to the Hausdorff measure in T with gauge function g(r)= r loglog1/r, restricted to T(a). As suggested by a result due to Perkins \citePer88,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∈ (0, ∞), ℓa(⋅) = (1)/(2) \mathscrHg ( ⋅ ∩ T(a)), where \mathscrHg stands for the g-Hausdorff measure.