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The stochastic wave equation in two spatial dimensions. (English) Zbl 0938.60046
Summary: 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.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
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