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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
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 $\mathbb{R}^d$.
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- Research Article
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- © EDP Sciences, SMAI, 2003
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