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

TL;DR

This paper investigates the expected perimeter length of the convex hull of hyperbolic Brownian motion, deriving asymptotics and exact formulas, revealing differences from Euclidean cases.

Contribution

It establishes a connection between hyperbolic hull perimeter expectations and exponential functionals of real Brownian motion, providing new asymptotic and exact results.

Findings

Large-time asymptotics are twice the Euclidean lower bound.

Exact perimeter expectation after exponential time is derived.

Asymptotic behavior differs from Euclidean Brownian motion with drift.

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.