Brownian Excursion Conditioned on Its Local Time

Electronic Communications in Probability Brownian Excursion Conditioned on Its Local Time

The Sde Solved by Local times of a Brownian Excursion or Bridge Derived from the Height Profile of a Random Tree or Forest1 by Jim Pitman

Ž .. Ž . v 4 on the interval 0, V X , where V X inf v: H X du t , and X 0 t t 0 u v Ž . for all v V X . This conditioned form of the Ray Knight description of t Brownian local times arises from study of the asymptotic distribution as ' n and 2k n l of the height profile of a uniform rooted random forest of k trees labeled by a set of n elements, as obtained by conditioning a uniform random mapp...

The SDE solved by local times of a Brownian excursion or bridge derived from the height pro le of a random tree or forest

Let B be a standard one-dimensional Brownian motion started at 0. Let Lt;v(jBj) be the occupation density of jBj at level v up to time t. The distribution of the process of local times (Lt;v(jBj); v 0) conditionally given Bt = 0 and Lt;0(jBj) = ` is shown to be that of the unique strong solution X of the Itô SDE dXv = n 4 X2 v t R v 0 Xudu 1 o dv + 2 p XvdBv on the interval [0; Vt(X)), where Vt...

A Vervaat-like Path Transformation for the Reeected Brownian Bridge Conditioned on Its Local Time at 0

We describe a Vervaat-like path transformation for the reeected 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 easi...

A Vervaat - like path transformation for the reflected Brownian bridge conditioned on its local time at 0

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 eas...