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Positivity of the density for the stochasticwave equation in two spatial dimensions

Published online by Cambridge University Press:  15 May 2003

Mireille Chaleyat–Maurel
Affiliation:
Université Pierre et Marie Curie, Laboratoire de Probabilités, 175/179 rue du Chevaleret, 75013 Paris, France; and Université René Descartes, 45 rue des Saints Pères, 75006 Paris, France; mcm@ccr.jussieu.fr.
Marta Sanz–Solé
Affiliation:
Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain; sanz@mat.ub.es.
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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 $\mathbb{R}^d$.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2003

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References

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.
Bally, V. and Pardoux, E., Malliavin Calculus for white-noise driven parabolic spde's. Potential Anal. 9 (1998) 27-64. CrossRef
Ben Arous, G. and Léandre, R., Décroissance exponentielle du noyau de la chaleur sur la diagonale (II). Probab. Theory Related Fields 90 (1991) 377-402.
Dalang, R. and Frangos, N., The stochastic wave equation in two spatial dimensions. Ann. Probab. 26 (1998) 187-212.
O. Lévêque, Hyperbolic stochastic partial differential equations driven by boundary noises. Thèse EPFL, Lausanne, 2452 (2001).
Márquez-Carreras, D., Mellouk, M. and Sarrà, M., On stochastic partial differential equations with spatially correlated noise: Smoothness of the law. Stochastic Proc. Appl. 93 (2001) 269-284. CrossRef
M. Métivier, Semimartingales. De Gruyter, Berlin (1982).
Millet, A. and Morien, P.-L., On a stochastic wave equation in two dimensions: Regularity of the solution and its density. Stochastic Proc. Appl. 86 (2000) 141-162. CrossRef
Millet, A. and Sanz-Solé, M., Points of positive density for the solution to a hyperbolic spde. Potential Anal. 7 (1997) 623-659. CrossRef
Millet, A. and Sanz-Solé, M., A stochastic wave equations in two space dimension: Smoothness of the law. Ann. Probab. 27 (1999) 803-844. CrossRef
Millet, A. and Sanz-Solé, M., Approximation and support theorem for a two space-dimensional wave equation. Bernoulli 6 (2000) 887-915. CrossRef
Morien, P.-L., Hölder and Besov regularity of the density for the solution of a white-noise driven parabolic spde. Bernoulli 5 (1999) 275-298. CrossRef
D. Nualart, Malliavin Calculus and Related Fields. Springer-Verlag (1995).
Nualart, D., 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.
Sanz-Solé, M. and Sarrà, M., 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.
Walsh, J.B., 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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