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Coverage probability of the planar Brownian convex hull
Abstract
Let 𝒦 be the convex hull of a planar Brownian motion up to time $1$. We explicitly determine the probability that 𝒦 contains a prescribed point z∈ℝ2. The proof is based on a general formula relating the radial function of a random convex body to its support function and its derivative, together with a comparison of 𝒦 with an arcsine-rescaled Gaussian ellipse. We also obtain an analogous formula for the convex hull of a planar Brownian bridge. Finally, we derive small- and large-time asymptotics for the corresponding coverage probabilities.
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BibTeX
@misc{kabluchko2026coverage,
title = {{Coverage probability of the planar Brownian convex hull}},
author = {Zakhar Kabluchko},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21687},
eprint = {2609.21687},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21687},
}
RIS
TY - UNPB
TI - Coverage probability of the planar Brownian convex hull
AU - Zakhar Kabluchko
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21687
UR - https://arxiv.org/abs/2609.21687
AB - Let 𝒦 be the convex hull of a planar Brownian motion up to time $1$. We explicitly determine the probability that 𝒦 contains a prescribed point z∈ℝ^2. The proof is based on a general formula relating the radial function of a random convex body to its support function and its derivative, together with a comparison of 𝒦 with an arcsine-rescaled Gaussian ellipse. We also obtain an analogous formula for the convex hull of a planar Brownian bridge. Finally, we derive small- and large-time asymptotics for the corresponding coverage probabilities.
ER -