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

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 $\frac{3π}{8}$ and $\fracπ{4}$ 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 $\mathcal{R}_ρ: = [ρ,\infty)\times{0}$ 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 $(x,0)\in \mathcal{R}_ρ$, the hitting time is distributed as the first passage time to the level $x$ of one-dimensional Brownian motion starting at $0$.

💡 Research Summary

The paper introduces and studies the “star hull” of planar Brownian motion and of the Brownian bridge. For a set A⊂ℝ² the star hull SH(A) is the smallest set that is star‑shaped with respect to the origin and contains A; equivalently it is obtained by filling in every line segment from the origin to a point of A. The authors first establish basic geometric facts: for a compact set K containing the origin the area of SH(K) can be expressed through the radial function r_K(θ) as
 area(SH(K)) = ½∫₀^{2π} r_K(θ)² dθ.
If a random continuous path X_t, 0≤t≤1, starts at the origin and is rotationally invariant, then the expected area of its star hull reduces to a single radial moment:
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