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 -
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