Convex hull of a Brownian motion in confinement

TL;DR

This paper investigates how confinement by a reflecting wall affects the average perimeter of the convex hull of a planar Brownian motion, revealing a surprising minimum and non-analytic behavior in the mean perimeter.

Contribution

It introduces a minimal model analyzing the impact of a reflecting wall on the convex hull of Brownian motion, highlighting non-monotonic and non-analytic effects.

Findings

Mean perimeter has a minimum with respect to initial distance to the wall.

Mean span exhibits non-analyticity at small distances.

Contrasts with the one-dimensional case where span is monotonic.

Abstract

We study the effect of confinement on the mean perimeter of the convex hull of a planar Brownian motion, defined as the minimum convex polygon enclosing the trajectory. We use a minimal model where an infinite reflecting wall confines the walk to its one side. We show that the mean perimeter displays a surprising minimum with respect to the starting distance to the wall and exhibits a non-analyticity for small distances. In addition, the mean span of the trajectory in a fixed direction {$\theta \in ]0,\pi/2[$}, which can be shown to yield the mean perimeter by integration over $\theta$, presents these same two characteristics. This is in striking contrast with the one dimensional case, where the mean span is an increasing analytical function. The non-monotonicity in the 2D case originates from the competition between two antagonistic effects due to the presence of the wall: reduction of the space accessible to the Brownian motion and effective repulsion.