## Un processus qui ressemble au pont Brownien. (A process resembling the Brownian bridge).(French)Zbl 0621.60086

Sémin. probabilités XXI, Lect. Notes Math. 1247, 270-275 (1987).
[For the entire collection see Zbl 0606.00022.]
Let $$(B_ t)$$ be a real Brownian motion starting at 0 and $$(\ell_ t,t\geq 0)$$ its local time at 0, $$\tau_ t=\inf \{u:\ell_ u>t\}$$. Put $$X_ u=B_{u\tau_ 1}/\sqrt{\tau_ 1}$$, $$p(u)=$$ Brownian bridge, and $$\lambda =$$ its local time at level 0 at time 1.
The main result asserts that, for each Borel functional $$F: C([0,1],{\mathbb{R}})\to {\mathbb{R}}_+$$, one has the identity $E(F(X_ u,0\leq u\leq 1))=E(F(p(u),0\leq u\leq 1)\sqrt{2/\pi}/\lambda).$
Reviewer: U.Krengel

### MSC:

 60J65 Brownian motion

### Keywords:

Brownian bridge; Brownian motion; local time

Zbl 0606.00022
Full Text: