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

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

### Citations:

Zbl 0619.60072; Zbl 0332.60055; Zbl 0551.60077