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Perimeter length of the convex hull of Brownian motion in the hyperbolic plane

Chinmoy Bhattacharjee, Rik Versendaal, Andrew Wade

TL;DR

This paper analyzes the perimeter of the convex hull of a Brownian path in the hyperbolic plane ${\mathbb H}^2$. It derives a fundamental link between the expected hyperbolic perimeter $\mathbb{E}L_t$ and a one-dimensional exponential functional of Brownian motion, showing $\mathbb{E}L_t = \sqrt{8\pi}\, \mathbb{E}\sqrt{\mathcal{E}_t}$ with $\mathcal{E}_t=\int_0^t e^{2W_s-s}\,ds$, and establishing small- and large-time asymptotics $\mathbb{E}L_t \sim \sqrt{8\pi t}$ and $\mathbb{E}L_t \sim 2t$, respectively. The authors exploit a Cauchy-type formula in hyperbolic geometry, a half-plane BM representation, and the theory of exponential functionals of Brownian motion (notably Hariya–Yor and Yor), to obtain an exact expression for $\mathbb{E}L_{T_\lambda}$ at independent Exp$(\lambda)$ times via Yor’s function $G$. They further prove a strong law for the radial coordinate and uniform convergence of the direction, revealing nuanced differences from Euclidean drifted Brownian motion and suggesting avenues for higher-dimensional hyperbolic spaces and distributional limit results. Overall, the work connects hyperbolic stochastic geometry with one-dimensional stochastic analysis to characterize the convex hull perimeter in a non-Euclidean setting and to reveal the delicate balance between angular contraction and radial growth.

Abstract

We relate the expected hyperbolic length of the perimeter of the convex hull of the trajectory of Brownian motion in the hyperbolic plane to an expectation of a certain exponential functional of a one-dimensional real-valued Brownian motion, and hence derive small- and large-time asymptotics for the expected hyperbolic perimeter. In contrast to the case of Euclidean Brownian motion with non-zero drift, the large-time asymptotics are a factor of two greater than the lower bound implied by the fact that the convex hull includes the hyperbolic line segment from the origin to the endpoint of the hyperbolic Brownian motion. We also obtain an exact expression for the expected perimeter length after an independent exponential random time.

Perimeter length of the convex hull of Brownian motion in the hyperbolic plane

TL;DR

This paper analyzes the perimeter of the convex hull of a Brownian path in the hyperbolic plane . It derives a fundamental link between the expected hyperbolic perimeter and a one-dimensional exponential functional of Brownian motion, showing with , and establishing small- and large-time asymptotics and , respectively. The authors exploit a Cauchy-type formula in hyperbolic geometry, a half-plane BM representation, and the theory of exponential functionals of Brownian motion (notably Hariya–Yor and Yor), to obtain an exact expression for at independent Exp times via Yor’s function . They further prove a strong law for the radial coordinate and uniform convergence of the direction, revealing nuanced differences from Euclidean drifted Brownian motion and suggesting avenues for higher-dimensional hyperbolic spaces and distributional limit results. Overall, the work connects hyperbolic stochastic geometry with one-dimensional stochastic analysis to characterize the convex hull perimeter in a non-Euclidean setting and to reveal the delicate balance between angular contraction and radial growth.

Abstract

We relate the expected hyperbolic length of the perimeter of the convex hull of the trajectory of Brownian motion in the hyperbolic plane to an expectation of a certain exponential functional of a one-dimensional real-valued Brownian motion, and hence derive small- and large-time asymptotics for the expected hyperbolic perimeter. In contrast to the case of Euclidean Brownian motion with non-zero drift, the large-time asymptotics are a factor of two greater than the lower bound implied by the fact that the convex hull includes the hyperbolic line segment from the origin to the endpoint of the hyperbolic Brownian motion. We also obtain an exact expression for the expected perimeter length after an independent exponential random time.

Paper Structure

This paper contains 17 sections, 10 theorems, 96 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Hyperbolic Brownian motion
  3. The convex hull and its perimeter length
  4. Discussion and open questions
  5. Background on hyperbolic geometry
  6. Models of the hyperbolic plane
  7. The Poincaré half plane model.
  8. The Poincaré disk model.
  9. The Beltrami--Klein disk.
  10. Hyperbolic Cauchy formula
  11. Hyperbolic Brownian motion on the Poincaré half-plane
  12. Overview
  13. Asymptotics for the half-plane process
  14. Exponential functionals of Brownian motion
  15. The expected perimeter and its asymptotics
  16. ...and 2 more sections

Key Result

Proposition 1.1

It holds that, a.s., Moreover, there exists a random variable $\theta_\infty$, with $\theta_\infty \sim \mathrm{Unif}\mkern2mu [0,2\pi)$, and, for every $s >0$, a random variable $\Theta_\infty^{(s)} \in \mathbb{R}$, such that Moreover, for every $s>0$,

Figures (1)

  • Figure 1: Simulated hyperbolic Brownian motion trajectory and its convex hull. The SDEs \ref{['eq:r-sde']} and \ref{['eq:theta-sde']} were each approximately solved, in turn, over time interval $[0,10]$ with an Euler scheme with $10^6$ steps: top left pane shows the $R$ process and top right pane shows the $\Theta^{(s)}$ process over time $[s,1]$ for $s = 10^{-3}$. The discrete approximation moving swiftly to a large negative value reflects the rapid spinning out from the origin. The bottom left pane shows the resulting Brownian trajectory $B_t$ (shown in red) over time $t \in [0,10]$ in a section of the Beltrami--Klein disk $\mathbb{D}_\mathrm{K}$; in this model geodesics are straight lines, so the convex hull $\mathcal{H}_{10}$ (boundary in blue) can be computed more easily. The bottom right pane shows that same trajectory (red) and convex hull (blue) represented in the Poincaré disk $\mathbb{D}_\mathrm{P}$. See §\ref{['sec:models']} for a summary of the different models of $\mathbb{H}^2$ and how one maps between them. Note that already by time $10$ the process appears very close to the boundary, so running longer simulations would yield little more visual information.

Theorems & Definitions (22)

  • Proposition 1.1
  • Remark 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Remark 1.5
  • Corollary 1.6
  • Example 2.1: Line segment
  • Remark 2.2
  • Theorem 3.1
  • Proposition 3.2
  • ...and 12 more