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): **

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

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

**Résumé:** We consider the random vector $u(t,\underline
x)=(u(t,x_1),\dots,u(t,x_d))$, where $t>0,\ x_1,\dots,x_d$ are distinct
points of $\R^2$
and $u$ denotes the stochastic process solution to a stochastic wave
equation driven by
a noise white in time and correlated in space. In a recent paper by Millet
and Sanz-Solé [8], sufficient conditions are given ensuring existence and
smoothness of
density for $u(t,\underline x)$. We study here the positivity of such
density. Using
techniques developped in [1] (see also [7]) based on
Analysis on an
abstract Wiener space, we characterize the set of points $y\in\R^d$ where
the density is
positive and we prove that, under suitable assumptions, the set is $\R^d$.

**Mots Clés:** *Stochastic partial differential equations ; Malliavin calculus ; wave
equation ; probability densities*

**Date:** 2000-10-05

**Prépublication numéro:** *PMA-614*