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

Hugo Panzo

TL;DR

This work analyzes the star hull of planar Brownian motion and Brownian bridge, establishing exact results for the star-hull areas via a detailed study of the first hitting time and place of a horizontal ray. A central tool is the joint Laplace transform $\mathbb{E}[e^{-\lambda T_\rho - \mu X_\rho}] = \mathrm{erfc}(\sqrt{\rho(\sqrt{2\lambda}+\mu)})$, together with a conditional structure: given the hitting place $X_\rho=x$, the hitting time $T_\rho$ is distributed as the first passage time to level $x$ of 1D Brownian motion. The paper further derives densities for $(T_\rho, X_\rho)$, the radial distances $R^{\mathrm{BM}}$ and $R^{\mathrm{BB}}$, and uses these to obtain the exact star-hull areas $\mathbb{E}[\mathrm{area}(\mathrm{SH}(\mathrm{BM}_{[0,1]}))]=\frac{3\pi}{8}$ and $\mathbb{E}[\mathrm{area}(\mathrm{SH}(\mathrm{BB}_{[0,1]}))]=\frac{\pi}{4}$, while providing upper bounds for the Brownian topological hull. The approach blends conformal invariance, Bessel-bridge functionals, and polar/hull geometry to connect stochastic hitting problems with geometric hull functionals, contributing new exact results and methods for planar Brownian hulls.

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 $\frac{3π}{8}$ 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 $0$.

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

TL;DR

This work analyzes the star hull of planar Brownian motion and Brownian bridge, establishing exact results for the star-hull areas via a detailed study of the first hitting time and place of a horizontal ray. A central tool is the joint Laplace transform , together with a conditional structure: given the hitting place , the hitting time is distributed as the first passage time to level of 1D Brownian motion. The paper further derives densities for , the radial distances and , and uses these to obtain the exact star-hull areas and , while providing upper bounds for the Brownian topological hull. The approach blends conformal invariance, Bessel-bridge functionals, and polar/hull geometry to connect stochastic hitting problems with geometric hull functionals, contributing new exact results and methods for planar Brownian hulls.

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 and 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 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 , the hitting time is distributed as the first passage time to the level of one-dimensional Brownian motion starting at .
Paper Structure (17 sections, 15 theorems, 72 equations)

This paper contains 17 sections, 15 theorems, 72 equations.

Table of Contents

  1. Introduction and main results
  2. Organization of the paper
  3. Three kinds of hulls
  4. Star hulls and radial functions
  5. Topological and convex hulls
  6. Probabilistic preliminaries
  7. Scaling and space-time transformations
  8. Conformal invariance and time change
  9. Quadratic functionals of Bessel bridges
  10. Complementary error function results
  11. Proofs of the main results
  12. Proof of Theorem \ref{['thm:joint_Laplace']}
  13. Proof of Theorem \ref{['thm:conditional_structure']}
  14. Proof of Theorem \ref{['thm:densities']}
  15. Proof of Theorem \ref{['thm:radial_distance']}
  16. ...and 2 more sections

Key Result

Theorem 1

Consider the planar Brownian bridge $(\boldsymbol{B}_t:0\leq t\leq 1)$ from $\boldsymbol{0}$ to $\boldsymbol{0}$, and let the corresponding topological hull $\mathcal{T}^{\mathrm{BB}}$, star hull $\mathcal{S}^{\mathrm{BB}}$, and convex hull $\mathcal{C}^{\mathrm{BB}}$ be defined as in eq:bridge_hull

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark 1
  • Theorem 6
  • Remark 2
  • Definition 1
  • Proposition 1
  • ...and 19 more