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

Mireille Chaleyat-Maurel; Marta Sanz-Solé

ESAIM: Probability and Statistics (2003)

- Volume: 7, page 89-114
- ISSN: 1292-8100

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topChaleyat-Maurel, Mireille, and Sanz-Solé, Marta. "Positivity of the density for the stochastic wave equation in two spatial dimensions." ESAIM: Probability and Statistics 7 (2003): 89-114. <http://eudml.org/doc/245850>.

@article{Chaleyat2003,

abstract = {We consider the random vector $u(t,\underline\{x\})=(u(t,x_1),\dots ,u(t,x_d))$, where $t>0,\ x_1,\dots ,x_d$ 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 $\mathbb \{R\}^d$.},

author = {Chaleyat-Maurel, Mireille, Sanz-Solé, Marta},

journal = {ESAIM: Probability and Statistics},

keywords = {stochastic partial differential equations; Malliavin calculus; wave equation; probability densities; positivity of the density; existence and smoothness of density; characterization results; abstract Wiener space},

language = {eng},

pages = {89-114},

publisher = {EDP-Sciences},

title = {Positivity of the density for the stochastic wave equation in two spatial dimensions},

url = {http://eudml.org/doc/245850},

volume = {7},

year = {2003},

}

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

PB - EDP-Sciences

VL - 7

SP - 89

EP - 114

AB - We consider the random vector $u(t,\underline{x})=(u(t,x_1),\dots ,u(t,x_d))$, where $t>0,\ x_1,\dots ,x_d$ 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 $\mathbb {R}^d$.

LA - eng

KW - stochastic partial differential equations; Malliavin calculus; wave equation; probability densities; positivity of the density; existence and smoothness of density; characterization results; abstract Wiener space

UR - http://eudml.org/doc/245850

ER -

## References

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