TITLE: Distribution of time spent above $0$ by a Brownian Bridge.
QUESTION [11 upvotes]: Let's say I have a Brownian motion, such that I know its value at time 0 (0) and time T (also 0). I am trying to evaluate the time spent above 0 between time 0 and T.
Obviously I know that the average of this value is 1/2, but is there a way to know the full law of that duration? Or are there other known theoretical results for this? I am guessing this is a fairly common problem but I am unable to find references for this.
Thanks!

REPLY [5 votes]: Yes the time spent above zero by the trajectory of a Brownian bridge  is a uniform random variable, see Exercise 3.9 in  Chapter 12 of  Continuous Martingales and Brownian Motion by Revuz and Yor.