Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

A stochastic wave equation in two space dimension : Smoothness of the law


Code(s) de Classification MSC:

Résumé: We prove the existence and uniqueness, for any time, of a real-valued process solving a non-linear stochastic wave equation driven by a Gaussian noise white in time and correlated in the two-dimensional space variable. We prove that the solution is regular in the sense of the Malliavin calculus. We also give a decay condition on the covariance function of the noise under which the solution has H\"{o}lder continuous trajectories and show that, under an additional ellipticity assumption, the law of the solution at any strictly positive time has a smooth density.

Mots Clés: Stochastic partial differential equation; wave equation; gaussian noise; Malliavin calculus; existence and smoothness of the density.

Date: 1997-09-25

Prépublication numéro: PMA-410