The Hausdorff measure of the boundary of the Brownian disk

TL;DR

This paper proves that the uniform measure on the boundary of the Brownian disk and half-plane coincides with a specific Hausdorff measure, linking the boundary's metric structure with its natural measure.

Contribution

It establishes the precise Hausdorff measure corresponding to the boundary of the Brownian disk and half-plane, showing the measure is determined by the boundary's metric.

Findings

Uniform measure on boundary matches Hausdorff measure with gauge function h(s)=κ s^2 log log(1/s)

Result applies to both Brownian disk and half-plane boundaries

Measures are uniquely determined by boundary metrics

Abstract

Consider the boundary $\partial \mathbb D$ 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 $\partial \mathbb D$ coincides with the Hausdorff measure associated with the gauge function $h(s)=\kappa s^2\log\log(1/s)$ for some deterministic constant $\kappa>0$. 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 $\mathbb H$).