Electronic Journal of Probability

Extending the Martingale Measure Stochastic Integral With Applications to Spatially Homogeneous S.P.D.E.'s

Robert Dalang

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We extend the definition of Walsh's martingale measure stochastic integral so as to be able to solve stochastic partial differential equations whose Green's function is not a function but a Schwartz distribution. This is the case for the wave equation in dimensions greater than two. Even when the integrand is a distribution, the value of our stochastic integral process is a real-valued martingale. We use this extended integral to recover necessary and sufficient conditions under which the linear wave equation driven by spatially homogeneous Gaussian noise has a process solution, and this in any spatial dimension. Under this condition, the non-linear three dimensional wave equation has a global solution. The same methods apply to the damped wave equation, to the heat equation and to various parabolic equations.

Article information

Electron. J. Probab. Volume 4 (1999), paper no. 6, 29 pp.

Accepted: 24 March 1999
First available in Project Euclid: 4 March 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 60H05: Stochastic integrals 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 35D10

stochastic wave equation stochastic heat equation Gaussian noise process solution

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Dalang, Robert. Extending the Martingale Measure Stochastic Integral With Applications to Spatially Homogeneous S.P.D.E.'s. Electron. J. Probab. 4 (1999), paper no. 6, 29 pp. doi:10.1214/EJP.v4-43. http://projecteuclid.org/euclid.ejp/1457125515.

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