The Hausdorff measure of the boundary of the Brownian disk

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 $\mathbb{H}$).

16 pages