Positivity of the density for the stochastic wave equation in two spatial dimensions
ESAIM: Probability and Statistics, Tome 7 (2003) , pp. 89-114.

We consider the random vector $u\left(t,\underline{x}\right)=\left(u\left(t,{x}_{1}\right),\cdots ,u\left(t,{x}_{d}\right)\right)$, where $t>0,\phantom{\rule{4pt}{0ex}}{x}_{1},\cdots ,{x}_{d}$ 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 $u\left(t,\underline{x}\right)$. 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 ${ℝ}^{d}$.

DOI : https://doi.org/10.1051/ps:2003002
Classification : 60H15,  60H07
Mots clés : 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/}
}
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. http://www.numdam.org/articles/10.1051/ps:2003002/

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