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The uniform random tree in a Brownian excursion. (English) Zbl 0794.60080

To any Brownian excursion \(e\) with duration \(\sigma(e)\) and any \(t_ 1, \dots,t_ p \in [0,\sigma (e)]\), we associate a branching tree with \(p\) branches denoted by \(T_ p (e, t_ 1, \dots, t_ p)\), which is closely related to the structure of the minima of \(e\). Our main theorem states that, if \(e\) is chosen according to the Itô measure and \((t_ 1, \dots, t_ p)\) according to the Lebesgue measure on \([0,\sigma (e)]^ p\), the tree \(T_ p (e,t_ 1, \dots, t_ p)\) is distributed according to the uniform measure on the set of trees with \(p\) branches. The proof of this result yields additional information about the “subexcursions” of \(e\) corresponding to the different branches of the tree, thus generalizing a well-known representation theorem of J.-M. Bismut [Z. Wahrscheinlichkeitstheorie Verw. Geb. 69, 65-98 (1985; Zbl 0551.60077)]. If we replace the Itô measure by the law of the normalized excursion, a simple conditioning argument leads to another remarkable result originally proved by D. J. Aldous [Ann. Probab. 19, No. 1, 1-28 (1991; Zbl 0722.60013), Stochastic analysis, Proc. Symp., Durham/UK 1990, Lond. Math. Soc. Lect. Note Ser. 167, 23-70 (1991; Zbl 0791.60008) and Ann. Probab. 21, No. 1, 248-289 (1993; Zbl 0791.60009)] with a very different method.

MSC:

60J65 Brownian motion
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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[1] [Ab] Abraham, R.: L’arbre aléatoire infini associé à une excursion brownienne. In: Azéma, J. et al. (eds.) Séminaire de Probabilités XXVI. (Lect. Notes Math., vol. 1526, pp. 374-397) Berlin Heidelberg New York: Springer 1992
[2] [A1] Aldous, D.J.: The continuum random tree I. Ann. Probab.19, 1-28 (1991) · Zbl 0722.60013 · doi:10.1214/aop/1176990534
[3] [A2] Aldous, D.J.: The continuum random tree II. In: Barlow, M.T., Bingham, N.H. (eds.) Stochastic analysis, pp. 23-70. Cambridge: Cambridge University Press 1991 · Zbl 0791.60008
[4] [A3] Aldous, D.J.: The continuum random tree III. Ann Probab. (to appear 1993) · Zbl 0791.60009
[5] [B] Bismut, J.M. Last exit decompositions and regularity at the boundary of transition probabilities. Z. Wahrscheinlichkeitstheor. Verw. Geb.69, 65-98 (1985) · Zbl 0551.60077 · doi:10.1007/BF00532586
[6] [B1] Blumenthal, R.M.: Excursions of Markov processes. Boston: Birkhäuser 1992 · Zbl 0983.60504
[7] [L1] Le Gall, J.F.: Une approche élémentaire des théorèmes de décomposition de Williams. In: Azéma, J., Yor, M. (eds.) Séminaire de Probabilités XX. (Lect. Notes Math., vol. 1204, pp. 447-464) Berlin Heidelberg New York: Springer 1986
[8] [L2] Le Gall, J.F.: Marches aléatoires, mouvement brownien et processus de branchement. In: Azéma, J. et al. (eds.) Séminaire de Probabilités XXIII. (Lect. Notes Math., vol. 1372, pp. 258-274) Berlin Heidelberg New York: Springer 1989 · Zbl 0741.60078
[9] [L3] Le Gall, J.F.: Brownian excursions, trees and measure-valued branching processes. Ann. Probab.19, 1399-1439 (1991) · Zbl 0753.60078 · doi:10.1214/aop/1176990218
[10] [NP1] Neveu, J., Pitman, J.W.: Renewal property of the extrema and tree property of a one-dimensional Brownian motion. In: Azéma, J. et al. (eds.) Séminaire de Probabilités XXIII. (Lect. Notes Math., vol. 1372, pp. 239-247) Berlin Heidelberg New York: Springer 1989 · Zbl 0741.60080
[11] [NP2] Neveu, J., Pitman, J.W.: The branching process in Brownian excursion. In: Azéma, J. et al. (eds.) Séminaire de Probabilités XXIII. (Lect. Notes Math., vol. 1372, pp. 248-257) Berlin Heidelberg New York: Springer 1989 · Zbl 0741.60081
[12] [RY] Revuz, D., Yor, M.: Continuous martingales and Brownian motion. Berlin Heidelberg New York: Springer 1991 · Zbl 0731.60002
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