The Annals of Probability

The stochastic wave equation in two spatial dimensions

Robert C. Dalang and N. E. Frangos

Full-text: Open access

Abstract

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.

Article information

Source
Ann. Probab. Volume 26, Number 1 (1998), 187-212.

Dates
First available: 31 May 2002

Permanent link to this document
http://projecteuclid.org/euclid.aop/1022855416

Mathematical Reviews number (MathSciNet)
MR1617046

Digital Object Identifier
doi:10.1214/aop/1022855416

Zentralblatt MATH identifier
0938.60046

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 35D10

Keywords
Stochastic wave equation Gaussian noise process solution

Citation

Dalang, Robert C.; Frangos, N. E. The stochastic wave equation in two spatial dimensions. The Annals of Probability 26 (1998), no. 1, 187--212. doi:10.1214/aop/1022855416. http://projecteuclid.org/euclid.aop/1022855416.


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