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April 1999 A Stochastic Wave Equation in Two Space Dimension: Smoothness of the Law
Annie Millet, Marta Sanz-Solé
Ann. Probab. 27(2): 803-844 (April 1999). DOI: 10.1214/aop/1022677387


We prove the existence and uniqueness, for any time, of a real-valued process solving a nonlinear stochastic wave equation driven by a Gaussian noise white in time and correlated in the two-dimensional space variable. We prove that the solution is regular in the sense of the Malliavin calculus. We also give a decay condition on the covariance function of the noise under which the solution has Hölder continuous trajectories and show that, under an additional ellipticity assumption, the law of the solution at any strictly positive time has a smooth density.


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Annie Millet. Marta Sanz-Solé. "A Stochastic Wave Equation in Two Space Dimension: Smoothness of the Law." Ann. Probab. 27 (2) 803 - 844, April 1999.


Published: April 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0944.60067
MathSciNet: MR1698971
Digital Object Identifier: 10.1214/aop/1022677387

Primary: 60H15
Secondary: 60H07

Keywords: existence and smoothness of the density , Gaussian noise , Malliavin calculus , Stochastic partial differential equation , wave equation

Rights: Copyright © 1999 Institute of Mathematical Statistics


Vol.27 • No. 2 • April 1999
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