Techniques biharmoniques pour l'étude du mouvement brownien de P. Lévy à trois paramètres. (Biharmonic techniques for the study of Lévy's three-parameter Brownian motion) (Q914245)

It is shown that a Lévy's Brownian motion indexed by \({\mathbb{R}}^ 3\), B(P), \(P\in {\mathbb{R}}^ 3\), can be decomposed as: \[ B(P)=V(P)+tZ(P)+Y(P),\quad P=(x,t)\in {\mathbb{R}}^ 2\times [0,\infty), \] in the sense of law, where V(P) is a centred Gaussian process with the Green function of the bi-Laplacian \(\Delta^ 2\) as covariance, the trajectories of Z(P) and Y(P) are harmonic in \({\mathbb{R}}^ 2\times (0,\infty)\) and biharmonic (i.e. the solutions of the equation \(\Delta^ 2u=0)\), respectively, and the three processes V(P), Z(P) and Y(P) are independent. The Markov property of V(P) is discussed, too. The results are also valid for Brownian motions indexed by \({\mathbb{R}}^{2k+1}\) (replace the bi-Laplacian \(\Delta^ 2\) by \(\Delta^{k+1})\).