Annals of Probability

The SDE Solved By Local Times of a Brownian Excursion or Bridge Derived From the Height Profile of a Random Tree or Forest

Jim Pitman

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Let $B$ be a standard one-dimensional Brownian motion started at 0. Let $L_{t,v}(|B|)$ be the occupation density of $|B|$ at level $v$ up to time $t$. The distribution of the process of local times $(L_{t,v}(|B|)v\geq0)$ conditionally given $B_{t}= 0$ and $L_{t,0}(|B|)=l$ is shown to be that of the unique strong solution $X$ of the Itô SDE, $$dX_v = \Big\{4 - X^2_v\Big(t - \int_0^v X_u\,du\Big)^{-1}\Big\}dv + 2\sqrt{X_v}\,dB_v$$ on the interval $[0,V_{t}(X))$, where $V_{t}(X):= \inf\{v:\int_0^v X_u,du=t\}$, and $X_v= 0$ for all $v\geq V_t(X)$. This conditioned from study of the Ray-Knight description of Brownian local times arises from study of the asymptotic distribution as $n\rightarrow\infty$ and $2k/\sqrt{n}\rightarrow l$ 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 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. For $l = 0$, corresponding to asymptotics of a uniform random tree, the SDE gives a description of the process of local times of a Brownian excursion which is equivalent to Jeulin’s description of these local times as a time change of twice a Brownian excursion. Another corollary is the Biane-Yor description of the local times of a reflecting Brownian ridge as a time-changed reversal of twice a Brownian meander of the same length.

Article information

Ann. Probab., Volume 27, Number 1 (1999), 261-283.

First available in Project Euclid: 29 May 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60J55: Local time and additive functionals 60J60: Diffusion processes [See also 58J65] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 05C05: Trees

Local time Bessel process Galton-Watson branching process Brownian meander Ray-Knight theorems random mapping


Pitman, Jim. The SDE Solved By Local Times of a Brownian Excursion or Bridge Derived From the Height Profile of a Random Tree or Forest. Ann. Probab. 27 (1999), no. 1, 261--283. doi:10.1214/aop/1022677262.

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