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

TL;DR

This paper calculates the expected areas of the star hulls of planar Brownian motion and bridge, revealing their probabilistic structure through hitting times and a simple Laplace transform.

Contribution

It provides explicit expected area formulas for the star hulls of planar Brownian motion and bridge, and introduces a novel probabilistic analysis of hitting times.

Findings

Expected area of Brownian motion star hull: 3π/8

Expected area of bridge star hull: π/4

Hitting time distribution relates to one-dimensional Brownian passage time

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 $0$.