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

Abstract

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) → R for t → ∞. We investigate how this limit is approached and obtain an exact analytical expression for the distribution of the fluctuations Δ(t) = [R − M x (t)]/R in the limit of large t in all dimensions. We find that the distribution of Δ(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 2πR with a stretched exponential decay 2 π R − ⟨ L ( t ) ⟩ ∝ R ( D t ) 1 / 4 e − 2 2 D t / 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.