# Distribution of time spent above $0$ by a Brownian Bridge.

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!

For a Brownian Bridge $X$ with $X_0=0=X_1$, the distribution is a Uniform distribution on $[0,1]$. This result is due to Levy. I found this in a book called Handbook of Beta distribution and its applications page 193.
Notice the Arcsin law for a Brownian motion says the amount of time spent by a Brownian motion above $0$ is Beta distribution with parameters $(1/2, 1/2)$ and uniform distribution is Beta distribution with parameter $(1,1)$.