Occupation times for single-file diffusion

We consider a file of identical Brownian particles moving on the same axis x'Ox without crossing each other. They all start from the origin O at time t = 0 and are stopped at some time t. Denoting by T the time spent on the half-line [Ox) by a given particle of the line, we establish analytical formulae for the first two moments < T > and < T-2 >. In particular, considering the limit of an infinite number of particles, we get, for the 'middle' particle <(T/t)(2)> = 3/8 + 1/8J(0)(pi) = 0.33696... (J(0) is a Bessel function). This result (and also numerical simulations) shows that the distribution of T, though being close to it, is not fully a constant one.