Expected area of the star hull of planar Brownian motion and bridge

Abstract

We study the star hull of planar Brownian motion and bridge. Roughly speaking, this is the smallest starshaped set (with respect to the origin) that contains the trace of the path. In particular, we prove that the expected areas of the star hulls are $\frac{3\pi}{8}$ and $\frac{\pi}{4}$ for planar Brownian motion and bridge, respectively. Our proofs rely on a detailed analysis of the first hitting time and place of a horizontal ray $\mathcal{R}_\rho : = [\rho,\infty)\times\{0\}$ by planar Brownian motion starting at the origin. After deriving a remarkably simple Laplace transform of this joint law, we uncover via a probabilistic argument a surprising conditional structure: conditionally on the first hitting place being the point $(x,0)\in \mathcal{R}_\rho$, the hitting time is distributed as the first passage time to the level $x$ of one-dimensional Brownian motion starting at $0$.