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.


60J65 Brownian motion
60G60 Random fields