## Un principe de réflexion pour le mouvement brownien de Paul Lévy à trois paramètres. (A reflection principle for the Lévy Brownian motion depending on three parameters).(French)Zbl 0634.60067

Let $$B(x)$$, $$x\in {\mathbb{R}}^ 3$$, be a three parameters Lévy Brownian motion. Denote by $$L(H^+)$$, $$L(H^-)$$ the closed subspaces of $$L^ 2$$ spanned respectively by $\{B(x): x\in H^+_ 0={\mathbb{R}}^ 2\times (0,+\infty)\}\quad and\quad \{B(x): x\in H^-_ 0={\mathbb{R}}^ 2\times (- \infty,0)\}.$ We prove that conditional on $$L(H^+)\cap L(H^-)$$ the processes B(x), $$x\in H^+_ 0$$, and $$B(y)$$, $$y\in H^-_ 0$$, are independent and identical in law. Moreover, the associated covariance function coincides with the Green function of the second kind for the bilaplacian $$\Delta^ 2$$ defined on $$H^+_ 0$$. Some consequences of this result are mentioned.

### MSC:

 60J65 Brownian motion 60G60 Random fields