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

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

- A. MILLET
**M. SANZ-SOLÉ**

**Code(s) de Classification MSC:**

- 60H15 Stochastic partial differential equations, See also {35R60}
- 60H07 Stochastic calculus of variations and the Malliavin calculus

**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*