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

Mireille Chaleyat–Maurel; Marta Sanz–Solé

ESAIM: Probability and Statistics (2010)

- 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 (2010): 89-114. <http://eudml.org/doc/104311>.

@article{Chaleyat2010,

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 $\mathbb\{R\}^d$.
},

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

journal = {ESAIM: Probability and Statistics},

keywords = {Stochastic partial differential equations;
Malliavin Calculus; wave
equation; probability densities.; 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},

month = {3},

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/104311},

volume = {7},

year = {2010},

}

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

DA - 2010/3//

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, 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 $\mathbb{R}^d$.

LA - eng

KW - Stochastic partial differential equations;
Malliavin Calculus; wave
equation; probability densities.; 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/104311

ER -

## References

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