Electronic Journal of Probability

Some Non-Linear S.P.D.E's That Are Second Order In Time

Robert Dalang and Carl Mueller

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We extend J.B. Walsh's theory of martingale measures in order to deal with stochastic partial differential equations that are second order in time, such as the wave equation and the beam equation, and driven by spatially homogeneous Gaussian noise. For such equations, the fundamental solution can be a distribution in the sense of Schwartz, which appears as an integrand in the reformulation of the s.p.d.e. as a stochastic integral equation. Our approach provides an alternative to the Hilbert space integrals of Hilbert-Schmidt operators. We give several examples, including the beam equation and the wave equation, with nonlinear multiplicative noise terms.

Article information

Electron. J. Probab. Volume 8 (2003), paper no. 1, 21 p.

First available in Project Euclid: 23 May 2016

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Mathematical Reviews number (MathSciNet)

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 35L05: Wave equation

Stochastic wave equation stochastic beam equation spatially homogeneous Gaussian noise stochastic partial differential equations


Dalang, Robert; Mueller, Carl. Some Non-Linear S.P.D.E's That Are Second Order In Time. Electron. J. Probab. 8 (2003), paper no. 1, 21 p. doi:10.1214/EJP.v8-123. https://projecteuclid.org/euclid.ejp/1464037574.

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