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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2411.16541 |
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| _version_ | 1866910715884863488 |
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| author | Metz--Donnadieu, Alexis |
| author_facet | Metz--Donnadieu, Alexis |
| contents | 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 $h(s)=κs^2\log\log(1/s)$ 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 $\mathbb H$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16541 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Hausdorff measure of the boundary of the Brownian disk Metz--Donnadieu, Alexis Probability 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 $h(s)=κs^2\log\log(1/s)$ 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 $\mathbb H$). |
| title | The Hausdorff measure of the boundary of the Brownian disk |
| topic | Probability |
| url | https://arxiv.org/abs/2411.16541 |