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

TL;DR

This paper investigates the distribution of the occupation time of the union of the first and third quadrants for planar Brownian motion, addressing a longstanding open problem in probability theory.

Contribution

It reformulates the open problem into an alternative setting involving standard Brownian motions and derives insights into the distribution of the occupation time.

Findings

Occupation time distribution relates to the arcsine law for adjacent quadrants.

The problem simplifies when considering two adjacent quadrants, following a known distribution.

Provides a new perspective on a classical open problem in stochastic processes.

Abstract

An open problem of interest, first infused into the applied probability community in the work of Bingham and Doney in 1988, (see \cite{Bingham}) is stated as follows: find the distribution of the quadrant occupation time of planar Brownian motion. In this short communication, we study an alternate formulation of this longstanding open problem: let $X(t), Y(t), t \geq 0$ be standard Brownian motions starting at $x,y$ respectively. Find the distribution of the total time $T=Leb\{t \in [0,1]: X(t) \times Y(t) >0\}$, 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.