Exact joint distributions of three global characteristic times for Brownian motion

We consider three global characteristic times for a one-dimensional Brownian motion $x(\tau)$ in the interval $\tau\in [0,t]$: the occupation time $t_{\rm o}$ denoting the cumulative time where $x(\tau)>0$, the time $t_{\rm m}$ at which the process achieves its global maximum in $[0,t]$ and the last-passage time $t_l$ through the origin before $t$. All three random variables have the same marginal distribution given by L\'evy's arcsine law. We compute exactly the pairwise joint distributions of these three times and show that they are quite different from each other. The joint distributions display rather rich and nontrivial correlations between these times. Our analytical results are verified by numerical simulations.