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

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

Positivity of the density for the stochastic wave equation in two spatial dimensions


Code(s) de Classification MSC:

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