The Annals of Probability
- Ann. Probab.
- Volume 26, Number 1 (1998), 187-212.
The stochastic wave equation in two spatial dimensions
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.
Ann. Probab., Volume 26, Number 1 (1998), 187-212.
First available in Project Euclid: 31 May 2002
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 35D10
Dalang, Robert C.; Frangos, N. E. The stochastic wave equation in two spatial dimensions. Ann. Probab. 26 (1998), no. 1, 187--212. doi:10.1214/aop/1022855416. https://projecteuclid.org/euclid.aop/1022855416