Statistics of the maximum and the convex hull of a Brownian motion in confined geometries

TL;DR

This paper analyzes the maximum displacement and convex hull of a Brownian particle in confined geometries, deriving exact distributions and exploring their dependence on dimension and boundary shape, supported by numerical simulations.

Contribution

It provides exact analytical expressions for the distribution of the maximum and convex hull properties of Brownian motion in confined domains, extending understanding across various geometries and dimensions.

Findings

Distribution of fluctuations depends on dimension

Convex hull perimeter converges slowly with stretched exponential decay

Results are validated by numerical simulations

Abstract

We consider a Brownian particle with diffusion coefficient $D$ in a $d$-dimensional ball of radius $R$ with reflecting boundaries. We study the maximum $M_x(t)$ of the trajectory of the particle along the $x$-direction at time $t$. In the long time limit, the maximum converges to the radius of the ball $M_x(t) \to R$ for $t\to \infty$. We investigate how this limit is approached and obtain an exact analytical expression for the distribution of the fluctuations $\Delta(t) = [R-M_x(t)]/R$ in the limit of large $t$ in all dimensions. We find that the distribution of $\Delta(t)$ exhibits a rich variety of behaviors depending on the dimension $d$. These results are obtained by establishing a connection between this problem and the narrow escape time problem. We apply our results in $d=2$ to study the convex hull of the trajectory of the particle in a disk of radius $R$ with reflecting boundaries. We find that the mean perimeter $\langle L(t)\rangle$ of the convex hull exhibits a slow convergence towards the perimeter of the circle $2\pi R$ with a stretched exponential decay $2\pi R-\langle L(t)\rangle \propto \sqrt{R}(Dt)^{1/4} \,e^{-2\sqrt{2Dt}/R}$. Finally, we generalise our results to other confining geometries, such as the ellipse with reflecting boundaries. Our results are corroborated by thorough numerical simulations.