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.
Categories: cond-mat.stat-mech