Exact joint distributions of three global characteristic times for Brownian motion

TL;DR

This paper derives exact joint distributions for three key global times in one-dimensional Brownian motion, revealing complex correlations and enriching understanding of their probabilistic structure.

Contribution

It provides the first exact joint distributions for occupation time, maximum time, and last-passage time in Brownian motion, highlighting their nontrivial correlations.

Findings

Joint distributions differ significantly from each other.

Distributions exhibit rich, nontrivial correlations.

Results confirmed by numerical simulations.

Abstract

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.