# 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

## Access Full Article

top## Abstract

top## How to cite

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

top- S. Aida, S. Kusuoka and D. Stroock, On the support of Wiener functionals, edited by K.D. Elworthy and N. Ikeda, Asymptotic Problems in Probability Theory: Wiener Functionals and Asymptotics. Longman Scient. and Tech., New York, Pitman Res. Notes in Math. Ser. 284 (1993) 3-34. Zbl0790.60047
- V. Bally and E. Pardoux, Malliavin Calculus for white-noise driven parabolic spde's. Potential Anal.9 (1998) 27-64. Zbl0928.60040
- G. Ben Arous and R. Léandre, Décroissance exponentielle du noyau de la chaleur sur la diagonale (II). Probab. Theory Related Fields90 (1991) 377-402.
- R. Dalang and N. Frangos, The stochastic wave equation in two spatial dimensions. Ann. Probab.26 (1998) 187-212. Zbl0938.60046
- O. Lévêque, Hyperbolic stochastic partial differential equations driven by boundary noises. Thèse EPFL, Lausanne, 2452 (2001).
- D. Márquez-Carreras, M. Mellouk and M. Sarrà, On stochastic partial differential equations with spatially correlated noise: Smoothness of the law. Stochastic Proc. Appl.93 (2001) 269-284. Zbl1053.60070
- M. Métivier, Semimartingales. De Gruyter, Berlin (1982).
- A. Millet and P.-L. Morien, On a stochastic wave equation in two dimensions: Regularity of the solution and its density. Stochastic Proc. Appl.86 (2000) 141-162. Zbl1028.60061
- A. Millet and M. Sanz-Solé, Points of positive density for the solution to a hyperbolic spde. Potential Anal.7 (1997) 623-659. Zbl0892.60059
- A. Millet and M. Sanz-Solé, A stochastic wave equations in two space dimension: Smoothness of the law. Ann. Probab.27 (1999) 803-844. Zbl0944.60067
- A. Millet and M. Sanz-Solé, Approximation and support theorem for a two space-dimensional wave equation. Bernoulli6 (2000) 887-915. Zbl0968.60059
- P.-L. Morien, Hölder and Besov regularity of the density for the solution of a white-noise driven parabolic spde. Bernoulli5 (1999) 275-298.
- D. Nualart, Malliavin Calculus and Related Fields. Springer-Verlag (1995). Zbl0837.60050
- D. Nualart, Analysis on the Wiener space and anticipating calculus, in École d'été de Probabilités de Saint-Flour. Springer-Verlag, Lecture Notes in Math.1690 (1998) 863-901.
- M. Sanz-Solé and M. Sarrà, Path properties of a class of Gaussian processes with applications to spde's, in Stochastic Processes, Physics and Geometry: New interplays, edited by F. Gesztesy et al. American Mathematical Society, CMS Conf. Proc.28 (2000) 303-316. Zbl0970.60057
- J.B. Walsh, An introduction to stochastic partial differential equations, in École d'été de Probabilités de Saint-Flour, edited by P.L. Hennequin. Springer-Verlag, Lecture Notes in Math.1180 (1986) 266-437.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.