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

TLDR

The wave equation in dimensions greater than two has a Green's function that is a Schwartz distribution rather than a function. The authors extend Walsh's martingale measure stochastic integral to handle such distributional Green's functions and use it to derive necessary and sufficient conditions for the existence of process solutions to the linear wave equation driven by spatially homogeneous Gaussian noise in any spatial dimension. The extended integral is applied to the wave, damped wave, heat, and various parabolic equations, providing a unified method for solving these SPDEs. They show that the stochastic integral process remains a real‑valued martingale even with distributional integrands, and that under the derived condition the nonlinear three‑dimensional wave equation admits a global solution.

Abstract

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.