How sojourn time distributions of Brownian motion are affected by different forms of conditioning

In this paper we study the distribution of the sojourn time $\Gamma_{t} = meas{s<t : B(s)>0}$, where $B(t)$, $t>0$ is a Brownian motion (with or without drift), under different conditions at an intermediate time $u\leqt$ (and possibly with an additional condition at time $t$). We obtain different forms of the arc-sine law, which display a “bell-shaped” structure (instead of the usual “U-shaped” classical density) when $(B(u)=0)$ is assumed. When the conditions $(B(u) = 0, B(t) < 0)$ or $(B(u) = 0, B(t) > 0)$ are taken into account, an asymmetrical bell-shaped density is obtained.