Abstract

We describe a Vervaat-like path transformation for the reflected Brownian bridge conditioned on its local time at 0: up to random shifts, this process equals the two processes constructed from a Brownian bridge and a Brownian excursion by adding a drift and then taking the excursions over the current minimum. As a consequence, these three processes have the same occupation measure, which is easily found. The three processes arise as limits, in three different ways, of profiles associated to hashing with linear probing, or, equivalently, to parking functions. 1 Introduction We regard the Brownian bridge b(t) and the normalized (positive) Brownian excursion e(t) as defined on the circle R=Z, or, equivalently, as defined on the whole real line, being periodic with period 1. We define, for a 0, the operator \Psi a on the set of bounded functions on the line by \Psi a f(t) = f(t) \Gamma at \Gamma inf \Gamma1!st (f(s) \Gamma as) = sup st (f(t) \Gamma f(s) \Gamma a(t \Gamma s))...

Citations