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

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 $⟨L(t)⟩$ of the convex hull exhibits a slow convergence towards the perimeter of the circle $2\pi R$ with a stretched exponential decay $2\pi R-⟨L(t)⟩\propto \sqrt{R}(Dt{)}^{1/4}{e}^{-2\sqrt{2Dt}/R}$. Finally, we generalise our results to other confining geometries, such as the ellipse with reflecting boundaries. Our results are corroborated by thorough numerical simulations.

arXiv comment19 pages, 7 figures