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

Dalang, Robert C.

Electron. J. Probab. 4, Paper No. 6, 29 p. (1999).

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)

Keywords:

stochastic wave equation; stochastic heat equation; Gaussian noise; process solution