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

We study the behaviour 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^(m)_N(t) which at least m out of N particles spend together in the disc within the time interval [0, t], and (iii) the time ^(m)_N(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 → ∞.