Trivial solutions for a non-lineartwo-space dimensional wave equation perturbed by space-time white noise

Abstract

Linear and nonlinear stochastic wave equations given by a space-time Gaussian white noise are considered in a space of dimension d≥2. In the linear case the solution is a random Schwartz distribution. In the nonlinear case existence and uniqueness of solutions is proven in the framework of Colombeau random generalized functions. In the case where the nonlinear drift is given by the Fourier transform ƒof a complex measure for instance when ƒ is a trigonometric function the solution is proven to be Lp-associated with the solution of the free equation, for any p ≥ 1