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Extending martingale measure stochastic integral with applications to spatially homogeneous S. P. D. E’s. (English) Zbl 0922.60056
Summary: 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 nonlinear 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.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H05 Stochastic integrals
35R60 PDEs with randomness, stochastic partial differential equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
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