Coverage probability of the planar Brownian convex hull

Abstract

Let $\mathcal{K}$ be the convex hull of a planar Brownian motion up to time $1$. We explicitly determine the probability that $\mathcal{K}$ contains a prescribed point $z\in {\mathbb{R}}^2$. 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 $\mathcal{K}$ 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.

16 pages