We consider the random vector $u(t,\underline{x})=(u(t,{x}_{1}),\cdots ,u(t,{x}_{d}))$, where $t>0,\phantom{\rule{4pt}{0ex}}{x}_{1},\cdots ,{x}_{d}$ 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 equations, Malliavin calculus, wave equation, probability densities

@article{PS_2003__7__89_0, 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://www.numdam.org/articles/10.1051/ps:2003002/} }

TY - JOUR AU - Chaleyat-Maurel, Mireille AU - Sanz-Solé, Marta TI - Positivity of the density for the stochastic wave equation in two spatial dimensions JO - ESAIM: Probability and Statistics PY - 2003 SP - 89 EP - 114 VL - 7 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2003002/ DO - 10.1051/ps:2003002 LA - en ID - PS_2003__7__89_0 ER -

%0 Journal Article %A Chaleyat-Maurel, Mireille %A Sanz-Solé, Marta %T Positivity of the density for the stochastic wave equation in two spatial dimensions %J ESAIM: Probability and Statistics %D 2003 %P 89-114 %V 7 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2003002/ %R 10.1051/ps:2003002 %G en %F PS_2003__7__89_0

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

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