Suppose we have a Wiener process $W$, and $U_x$ is the amount of time spent below the level of $x$ during the time interval $(0,1)$. How can I calculate the probability density function of $U_x$. Does this has anything to do with local time or arc sine law?

Stephan, could you put it in more details please? Is it something like $\frac{1}{\pi\sqrt{u(1-u)}}e^{-\frac{x^2}{2u}}$?