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

We study the star hull of planar Brownian motion and bridge, and relate this random compact set to the more familiar convex and topological hulls.

Roughly speaking, the star hull 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.

Along the way, we find the one-point marginal distribution of the radial functions of both traces.

Our proofs rely on a detailed analysis of the first hitting time and place of a half-line by planar Brownian motion, and one of our main results is a remarkably simple expression for the Laplace transform of this joint law.