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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  1. 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.  
  2. V. Bally and E. Pardoux, Malliavin Calculus for white-noise driven parabolic spde's. Potential Anal.9 (1998) 27-64.  
  3. G. Ben Arous and R. Léandre, Décroissance exponentielle du noyau de la chaleur sur la diagonale (II). Probab. Theory Related Fields90 (1991) 377-402.  
  4. R. Dalang and N. Frangos, The stochastic wave equation in two spatial dimensions. Ann. Probab.26 (1998) 187-212.  
  5. O. Lévêque, Hyperbolic stochastic partial differential equations driven by boundary noises. Thèse EPFL, Lausanne, 2452 (2001).  
  6. 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.  
  7. M. Métivier, Semimartingales. De Gruyter, Berlin (1982).  
  8. 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.  
  9. A. Millet and M. Sanz-Solé, Points of positive density for the solution to a hyperbolic spde. Potential Anal.7 (1997) 623-659.  
  10. A. Millet and M. Sanz-Solé, A stochastic wave equations in two space dimension: Smoothness of the law. Ann. Probab.27 (1999) 803-844.  
  11. A. Millet and M. Sanz-Solé, Approximation and support theorem for a two space-dimensional wave equation. Bernoulli6 (2000) 887-915.  
  12. P.-L. Morien, Hölder and Besov regularity of the density for the solution of a white-noise driven parabolic spde. Bernoulli5 (1999) 275-298.  
  13. D. Nualart, Malliavin Calculus and Related Fields. Springer-Verlag (1995).  
  14. 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.  
  15. 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.  
  16. 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.  

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