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(X) := inffv : R v 0 Xudu = tg, and Xv = 0 for all v Vt(X). This conditioned form of the Ray-Knight description of Brownian local times arises from study of the asymptotic distribution as n!1 and 2k=pn! ` of the height pro le of a uniform rooted random forest of k trees labeled by a set of n elements, as obtained by conditioning a uniform random mapping of the set to itself to have k cyclic points. The SDE is the continuous analog of a simple description of a Galton-Watson branching process conditioned on its total progeny. A result is obtained regarding the weak convergence of normalizations of such conditioned Galton-Watson processes and height pro les of random forests to a solution of the SDE. For ` = 0, corresponding to asymptotics of a uniform random tree, the SDE gives a new description of the process of local times of a Brownian excursion, implying Jeulin's description of these local times as a time change of twice a Brownian excursion. Another corollary is the Biane-Yor Research supported in part by N.S.F. Grant DMS97-03961