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COMMUNICATIONS in PROBABILITY BROWNIAN EXCURSION CONDITIONED ON ITS LOCAL TIME

time. 1

Elect. Comm. in Probab. 3 (1998) 79{90

ELECTRONIC COMMUNICATIONS in PROBABILITY

BROWNIAN EXCURSION CONDITIONED ON ITS LOCAL TIME e-mail: aldous@stat.berkeley.edu http://www.stat.berkeley.edu/users/aldous

Department of Statistics University of California Berkeley CA 94720

DAVID J. ALDOUS1

AMS 1991 Subject classi cation: 60J55, 60J65, 60C05. Keywords and phrases: Brownian excursion, continuum random tree, Kingman's coalescent, local time.

submitted April 11, 1998; revised September 22nd, 1998

1 Introduction Let (Bu; 0 u 1) be standard Brownian excursion and (Ls; 0 precisely its local time at time 1: Z

s< 1) its local time, more

h

0

Ls ds=

Z 1

0

1(Bu h) du; h 0:

Biane - Yor 4] give an extensive treatment, including an elegant description of the law of L as a random time-change of the Brownian excursion: d ( 1 Ls=2; s 0)= (B 2 ?1 (s); s

0) for (t)=

Z

t

0

1=Bs ds

d where= indicates equality in law. Takacs 14] gives a combinatorial approach to formulas for the marginal law of Ls . Bertoin - Pitman 3] discuss transformations between Brownian excursion and other Brownian-type processes. References to further papers on standard Brownian excursion can be found in those references. Consider the question` Given a function`= (`(s); 0 s< 1), can we de ne a process B`= (Bu; 0 u 1) whose law (`) is, in some sense, the conditional law of B given L=`? As discussed in section 1.1, Warren and Yor 16] have recently given a quite di erent analysis of a similar question, and related ideas appeared earlier in the superprocesses literature. Of course, 1 Research supported by N.S.F. Grant DMS96-22859

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