Let 
$$X\_t = m \cdot t + W\_t$$
where $W\_t$ is a Brownian motion. Let
$$Z = \sup \{ t\in [0,1] : X\_t = 0\}.$$
It is known that if $m = 0$ then the distribution of $z$ is given by
$$\mathbb{P}[Z \leq y ] = \frac{2}{\pi} \arcsin \left( \sqrt{ y } \right).$$


What is the distribution of $z$ for general values of $m$?



Arcsine law for Brownian motion with drift
Here is a simple description:


$$Z\stackrel{d}{=}A\cdot Y\_m$$ i.e. $Z$ is distributed as the product of two random variables, where the the factors are independent, $A$ is $\arcsin$-distributed, and $Y\_m$ is distributed as $\min\{1,\exp(m^2/2)\}$.  

($\exp(\lambda)$ denotes an exponentially distributed rv. with exp. $1/\lambda$)


This can be proved using the random walk approximation of BM with drift which was used by Tak'acs (*On a generalization of the arc-sine law*, Ann. Appl. Prob., $\mathbf{3}$ (1996), 1035-1040.)
to study the sojourn time. (In that paper he gave an expression for the distribution of the sojourn time of BM with drift, which was much simpler than the original
solution by Akahori). In a similar way one can solve the corresponding random walk problem combinatorially and pass to the limit. 


