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Positivity of the density for the stochasticwave equation in two spatial dimensions

Published online by Cambridge University Press:  15 May 2003

Mireille Chaleyat–Maurel  and
Marta Sanz–Solé
Mireille Chaleyat–Maurel
Affiliation:
Université Pierre et Marie Curie, Laboratoire de Probabilités, 175/179 rue du Chevaleret, 75013 Paris, France; and Université René Descartes, 45 rue des Saints Pères, 75006 Paris, France; mcm@ccr.jussieu.fr.
Marta Sanz–Solé
Affiliation:
Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain; sanz@mat.ub.es.
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Abstract

We consider the random vector $u(t,\underline x)=(u(t,x_1),\dots,u(t,x_d))$, where t > 0, x1,...,xd are distinct points of $\mathbb{R}^2$ 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 [10], 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 [9]) based on Analysis on an abstract Wiener space, we characterize the set of points $y\in\mathbb{R}^d$ where the density is positive and we prove that, under suitable assumptions, this set is $\mathbb{R}^d$.

Keywords

Stochastic partial differential equationsMalliavin Calculuswave equationprobability densities.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 7 , March 2003 , pp. 89 - 114
Copyright
© EDP Sciences, SMAI, 2003

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