The Annals of Probability

Second-order linear hyperbolic SPDEs driven by isotropic Gaussian noise on a sphere

Robert C. Dalang and Olivier Lévêque

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We study a class of linear hyperbolic stochastic partial differential equations in bounded domains, which includes the wave equation and the telegraph equation, driven by Gaussian noise that is white in time but not in space. We give necessary and sufficient conditions on the spatial correlation of the noise for the existence (and uniqueness) of square-integrable solutions. In the particular case where the domain is a ball and the noise is concentrated on a sphere, we characterize the isotropic Gaussian noises with this property. We also give explicit necessary and sufficient conditions when the domain is a hypercube and the Gaussian noise is concentrated on a hyperplane.

Article information

Ann. Probab., Volume 32, Number 1B (2004), 1068-1099.

First available in Project Euclid: 11 March 2004

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Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 60G15: Gaussian processes 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]

Stochastic partial differential equations isotropic Gaussian noise hyperbolic equations


Dalang, Robert C.; Lévêque, Olivier. Second-order linear hyperbolic SPDEs driven by isotropic Gaussian noise on a sphere. Ann. Probab. 32 (2004), no. 1B, 1068--1099. doi:10.1214/aop/1079021472.

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