Open Access
January 1998 The stochastic wave equation in two spatial dimensions
Robert C. Dalang, N. E. Frangos
Ann. Probab. 26(1): 187-212 (January 1998). DOI: 10.1214/aop/1022855416


We consider the wave equation in two spatial dimensions driven by space–time Gaussian noise that is white in time but has a nondegenerate spatial covariance. We give a necessary and sufficient integral condition on the covariance function of the noise for the solution to the linear form of the equation to be a real-valued stochastic process, rather than a distribution-valued random variable. When this condition is satisfied, we show that not only the linear form of the equation, but also nonlinear versions, have a real-valued process solution. We give stronger sufficient conditions on the spatial covariance for the solution of the linear equation to be continuous, and we provide an estimate of its modulus of continuity.


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Robert C. Dalang. N. E. Frangos. "The stochastic wave equation in two spatial dimensions." Ann. Probab. 26 (1) 187 - 212, January 1998.


Published: January 1998
First available in Project Euclid: 31 May 2002

zbMATH: 0938.60046
MathSciNet: MR1617046
Digital Object Identifier: 10.1214/aop/1022855416

Primary: 60H15
Secondary: 35D10 , 35R60

Keywords: Gaussian noise , process solution , Stochastic wave equation

Rights: Copyright © 1998 Institute of Mathematical Statistics

Vol.26 • No. 1 • January 1998
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