Mean perimeter and area of the convex hull of a planar Brownian motion in the presence of resetting

TL;DR

This paper provides exact formulas for the mean perimeter and area of the convex hull of a 2D Brownian motion with resetting, revealing slow logarithmic growth and a late-time circular shape.

Contribution

It derives explicit expressions for the mean perimeter and area of the convex hull under resetting, a novel analytical result in stochastic geometry.

Findings

Mean perimeter scales as ln(rt) for large t

Mean area scales as ln^2(rt) for large t

Convex hull approaches a circular shape at late times

Abstract

We compute exactly the mean perimeter and the mean area of the convex hull of a $2$-d Brownian motion of duration $t$ and diffusion constant $D$, in the presence of resetting to the origin at a constant rate $r$. We show that for any $t$, the mean perimeter is given by $\langle L(t)\rangle= 2 \pi \sqrt{\frac{D}{r}}\, f_1(rt)$ and the mean area is given by $\langle A(t) \rangle= 2\pi\frac{D}{r}\, f_2(rt)$ where the scaling functions $f_1(z)$ and $f_2(z)$ are computed explicitly. For large $t\gg 1/r$, the mean perimeter grows extremely slowly as $\langle L(t)\rangle \propto \ln (rt)$ with time. Likewise, the mean area also grows slowly as $\langle A(t)\rangle \propto \ln^2(rt)$ for $t\gg 1/r$. Our exact results indicate that the convex hull, in the presence of resetting, approaches a circular shape at late times. Numerical simulations are in perfect agreement with our analytical predictions.