The Hausdorff measure of the boundary of the Brownian disk

2024/11/25 by Alexis Metz–Donnadieu, Metz--Donnadieu, Alexis
Mathematics · #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #Probability (math.PR) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2411.16541

openalex publication_date 2024/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider the boundary ∂ \mathbb D of the Brownian disk \mathbb D as a metric space by endowing it with the (restriction of the) metric of \mathbb D. We show that the uniform measure on ∂ \mathbb D coincides with the Hausdorff measure associated with the gauge function h(s)=κs²loglog(1/s) for some deterministic constant κ>0. We also state the analogous result for the boundary of the Brownian half-plane \mathbb H. This proves in particular that the uniform measure on the boundary of the Brownian disk (resp. the Brownian half-plane) is determined by the metric on \mathbb D (resp. on \mathbb H).

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