An Extension of Vervaats Transformation and Its Consequences

Vervaat(18) proved that by exchanging the pre-minimum and post-minimum parts of a Brownian bridge one obtains a normalized Brownian excursion. Let s (element of) (0, 1), then we extend this result by determining a random time m s such that when we exchange the pre-m s-part and the post-m s-part of a Brownian bridge, one gets a Brownian bridge conditioned to spend a time equal to s under 0. This transformation leads to some independence relations between some functionals of the Brownian bridge and the time it spends under 0. By splitting the Brownian motion at time m s in another manner, we get a new path transformation which explains an identity in law on quantiles due to Port. It also yields a pathwise construction of a Brownian bridge conditioned to spend a time equal to s under 0.