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

Abstract

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We consider the random vector u ( t , x ̲ ) = ( u ( t , x 1 ) , , u ( t , x d ) ) , where t > 0, x1,...,xd are distinct points of 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 , 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 d where the density is positive and we prove that, under suitable assumptions, this set is d .

How to cite

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Chaleyat–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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