On the joint residence time of N independent two-dimensional Brownian motions

TL;DR

This paper derives simple exact formulas for the expected joint residence times of multiple independent two-dimensional Brownian particles within a disc, considering various scenarios of simultaneous occupancy.

Contribution

It provides the first explicit formulas for the expectations of joint residence times of N independent Brownian motions in a 2D disc, including cases of all, at least m, and exactly m particles.

Findings

Explicit formulas for expectations of joint residence times in the limit t→∞.

Analysis of different occupancy scenarios among particles.

Simplification of complex stochastic residence time calculations.

Abstract

We study the behavior of several joint residence times of N independent Brownian particles in a disc of radius $R$ in two dimensions. We consider: (i) the time T_N(t) spent by all N particles simultaneously in the disc within the time interval [0,t]; (ii) the time T_N^{(m)}(t) which at least m out of N particles spend together in the disc within the time interval [0,t]; and (iii) the time {\tilde T}_N^{(m)}(t) which exactly m out of N particles spend together in the disc within the time interval [0,t]. We obtain very simple exact expressions for the expectations of these three residence times in the limit t\to\infty.