An extension of Vervaat's transformation and its consequences

Vervaat((18)) proved that by exchanging the pre-minimum and post-minimum pa rts of a Brownian bridge one obtains a normalized Brownian excursion. Let s epsilon (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 Bets a Brownian bridge conditioned to spend a time e qual to s under 0. This transformation leads to some independence relations between some functionals of the Brownian bridge and the time it spends und er 0. By splitting the Brownian motion at time iii, in another manner, we g el 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 c onditioned to spend a time equal to s under 0.