Positivity of the density for the stochastic wave equation in two spatial dimensions
ESAIM: Probability and Statistics, Tome 7 (2003), pp. 89-114.

We consider the random vector u(t,x ̲)=(u(t,x 1 ),,u(t,x d )), where t>0,x 1 ,,x d 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 .

DOI : 10.1051/ps:2003002
Classification : 60H15, 60H07
Mots clés : stochastic partial differential equations, Malliavin calculus, wave equation, probability densities
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Chaleyat-Maurel, Mireille; Sanz-Solé, Marta. Positivity of the density for the stochastic wave equation in two spatial dimensions. ESAIM: Probability and Statistics, Tome 7 (2003), pp. 89-114. doi : 10.1051/ps:2003002. http://www.numdam.org/articles/10.1051/ps:2003002/

[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. | MR | Zbl

[2] V. Bally and E. Pardoux, Malliavin Calculus for white-noise driven parabolic spde's. Potential Anal. 9 (1998) 27-64. | Zbl

[3] G. Ben Arous and R. Léandre, Décroissance exponentielle du noyau de la chaleur sur la diagonale (II). Probab. Theory Related Fields 90 (1991) 377-402. | MR | Zbl

[4] R. Dalang and N. Frangos, The stochastic wave equation in two spatial dimensions. Ann. Probab. 26 (1998) 187-212. | MR | Zbl

[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. | Zbl

[7] M. Métivier, Semimartingales. De Gruyter, Berlin (1982). | MR | Zbl

[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. | MR | Zbl

[9] A. Millet and M. Sanz-Solé, Points of positive density for the solution to a hyperbolic spde. Potential Anal. 7 (1997) 623-659. | Zbl

[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. | Zbl

[11] A. Millet and M. Sanz-Solé, Approximation and support theorem for a two space-dimensional wave equation. Bernoulli 6 (2000) 887-915. | Zbl

[12] P.-L. Morien, Hölder and Besov regularity of the density for the solution of a white-noise driven parabolic spde. Bernoulli 5 (1999) 275-298. | MR | Zbl

[13] D. Nualart, Malliavin Calculus and Related Fields. Springer-Verlag (1995). | MR

[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. | Zbl

[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. | Zbl

[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. | Zbl

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