How the 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[less-than-or-equals, slant]t (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)[greater, less]0) are taken into account, an asymmetrical bell-shaped density is obtained.