Exact joint distributions of three global characteristic times for Brownian motion.

Abstract

We consider three global chracteristic times for a one-dimensional Brownian motion x(τ) in the interval τ∈[0,t]: the occupation time t_{o} denoting the cumulative time where x(τ)>0, the time t_{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 Lévy'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.