On occupation times of the first and third quadrants for planar Brownian motion

In Bingham and Doney (1988) the authors presented the applied probability community with a question which is very simply stated, yet is extremely difficult to solve: what is the distribution of the quadrant occupation time of planar Brownian motion? In this paper we study an alternate formulation of this long-standing open problem: let X(t), Y(t) t≥0, be standard Brownian motions starting at x, y, respectively. Find the distribution of the total time T=Leb{t∈[0,1]: X(t)×Y(t)>0}, when x=y=0, i.e. the occupation time of the union of the first and third quadrants. If two adjacent quadrants are used, the problem becomes much easier and the distribution of T follows the arcsine law.