Quelques précisions sur le méandre brownien. (Some precisions on the Brownian meander). (French) Zbl 0665.60081

Biane, Ph.; Yor, M.

Bull. Sci. Math., II. Sér. 112, No. 1, 101-109 (1988).

Let \(M^ t\) (resp. \(R^ t)\) be the law of the Brownian meander (resp. of the Bessel process of dimension 3) on the time interval [0,t]. By means of a previous result of the authors [ibid. 111, 23-101 (1987; Zbl 0619.60072)] they give simple proofs of the absolute continuity of \(M^ t\) with respect to \(R^ t\) and of a representation of \(M^ t\) by means of the law of the Brownian motion. Moreover they show that \((| \beta_ t| +\lambda_ t\), \(\lambda_ t\); \(0\leq t\leq 1)\), where \(\beta_ t\) is the Brownian bridge and \(\lambda\) its local time in zero, has the same law as \((m_ t, j_ t\); \(0\leq t\leq 1)\), where m is the Brownian meander and \(j_ t=\inf \{m_ s\); \(t\leq s\leq 1\}.\)
A similar result for Brownian motion and the Bessel process of dimension 3 has been given by J. W. Pitman [Adv. Appl. Probab. 7, 511-526 (1975; Zbl 0332.60055)]. Finally the authors give a simple proof of a decomposition of the Brownian path at its maximum on the interval [0,t] and of an associated asymptotic result, using a theorem of J. M. Bismut [Z. Wahrscheinlichkeitstheor. Verw. Geb. 69, 65-98 (1985; Zbl 0551.60077)].

Reviewer: M.Dozzi

MSC:

60J65

Brownian motion

Keywords:

Brownian meander; Bessel process; Brownian bridge; decomposition of the Brownian path

Citations:

Zbl 0619.60072; Zbl 0332.60055; Zbl 0551.60077