Stirring two grains of sand

TL;DR

This paper studies the behavior of two reflecting Brownian particles in a large torus, showing they become independent in distribution as the torus size grows infinitely large.

Contribution

It proves the convergence of the joint distribution of two reflecting Brownian particles to independent Brownian motions on an expanding torus, with dimension-dependent diffusion coefficients.

Findings

Joint distribution converges to independent Brownian motions as r→∞

Rescaled stationary distributions become product of uniform distributions

Diffusion coefficient depends on the dimension

Abstract

Consider two unit balls in a $d$-dimensional flat torus with edge length $r$, for $d\geq 2$. The balls do not move by themselves but they are pushed by a Brownian motion. The balls never intersect---they reflect if they touch. It is proved that the joint distribution of the processes representing the centers of the balls converges to the distribution of two independent Brownian motions when $r\to \infty$, assuming that we use a proper clock and proper scaling. The diffusion coefficient of the limit process depends on the dimension. The positions of the balls are asymptotically independent also in the following sense. The rescaled stationary distributions of the centers of the balls converge to the product of the stationary (hence uniform) distributions for each ball separately, as $r\to\infty$.