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

Chaleyat-Maurel, Mireille; Sanz-Solé, Marta

doi:10.1051/ps:2003002

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

Chaleyat-Maurel, Mireille ; Sanz-Solé, Marta

ESAIM: Probability and Statistics, Tome 7 (2003), pp. 89-114

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Résumé

We consider the random vector $u (t, \underset{̲}{x}) = (u (t, x_{1}), \dots, u (t, x_{d}))$ , 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 [10], sufficient conditions are given ensuring existence and smoothness of density for $u (t, \underset{̲}{x})$ . 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 $y \in ℝ^{d}$ where the density is positive and we prove that, under suitable assumptions, this set is $ℝ^{d}$ .

Zbl

DOI : 10.1051/ps:2003002

Classification : 60H15, 60H07
Keywords: stochastic partial differential equations, Malliavin calculus, wave equation, probability densities

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     author = {Chaleyat-Maurel, Mireille and Sanz-Sol\'e, Marta},
     title = {Positivity of the density for the stochastic wave equation in two spatial dimensions},
     journal = {ESAIM: Probability and Statistics},
     pages = {89--114},
     publisher = {EDP-Sciences},
     volume = {7},
     year = {2003},
     doi = {10.1051/ps:2003002},
     zbl = {1021.60049},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/ps:2003002/}
}

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Chaleyat-Maurel, Mireille; Sanz-Solé, Marta. Positivity of the density for the stochastic wave equation in two spatial dimensions. ESAIM: Probability and Statistics, Tome 7 (2003), pp. 89-114. doi: 10.1051/ps:2003002

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