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February 2004 A stochastic wave equation in dimension 3: smoothness of the law
Lluís Quer-Sardanyons, Marta Sanz-Solé
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Bernoulli 10(1): 165-186 (February 2004). DOI: 10.3150/bj/1077544607


We prove the existence and regularity of the density of the real-valued solution to a three-dimensional stochastic wave equation. The noise is white in time and has a spatially homogeneous correlation whose spectral measure μ satisfies R 3 μ(dξ)( 1+|ξ| 2) - η< , for some η (0,1 2 ) . Our approach uses the mild formulation of the equation given by means of Dalang's extended version of Walsh's stochastic integration. We apply the tools of Malliavin calculus on the appropriate Gaussian space related to the noise. An extension of Dalang's stochastic integral to the Hilbert-valued setting is needed. Let S3 be the fundamental solution to the three-dimensional wave equation. The assumption on the noise yields upper and lower bounds for the integral 0 t ds R 3 μ(dξ)| cal F S 3(s)(ξ)| 2 and upper bounds for 0 t ds R 3 μ(dξ)|ξ|| cal F S 3(s)(ξ)| 2 in terms of powers of t. These estimates, together with a suitable mollifying procedure for S3, are crucial in the analysis of the inverse of the Malliavin variance.


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Lluís Quer-Sardanyons. Marta Sanz-Solé. "A stochastic wave equation in dimension 3: smoothness of the law." Bernoulli 10 (1) 165 - 186, February 2004.


Published: February 2004
First available in Project Euclid: 23 February 2004

zbMATH: 1045.60068
MathSciNet: MR2044597
Digital Object Identifier: 10.3150/bj/1077544607

Rights: Copyright © 2004 Bernoulli Society for Mathematical Statistics and Probability


Vol.10 • No. 1 • February 2004
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