Brownian excursion conditioned on its local time
Let us consider \(l:[0,\infty[\rightarrow[0,\infty[\) a Borel measurable function satisfying some additional conditions. The author gives a construction of a process \((B^l_{u},0\leq u\leq 1)\), taking values in the set of positive continuous excursions on \([0,1]\), whose local time is \( l\). We denote by \(\psi(l)\) the law of \(B^l\), by \((B_{u}, 0\leq u\leq 1)\) a standard Brownian excursion and by \(L\) its local time. Then \(\psi(l)\) is a version of the conditional law of \(B\) given \(L= l\) [see also \textit{J. Warren} and \textit{M. Yor}, in: Séminaire de probabilités XXXII. Lect. Notes Math. 1686, 328-342 (1998) for a similar question, and \textit{J.-F. Le Gall}, Ann. Probab. 19, No. 4, 1399-1439 (1991; Zbl 0753.60078) for some conceptually related results in the setting of super-processes]. The construction needs a careful treatment of the correspondence between an excursion function and trees [see \textit{D. J. Aldous}, ibid. 21, No. 1, 248-289 (1993; Zbl 0791.60009) for the general theory] and a non-homogeneous analog of Kingman's coalescent [see \textit{J. F. C. Kingman}, in: Exchangeability in probability and statistics, 97-112 (1982; Zbl 0494.92011) and Stochastic Processes Appl. 13, 235-248 (1982; Zbl 0491.60076)].
scientific article; zbMATH DE number 1228297