Nonlinear SPDEs: Colombeau Solutions and Pathwise Limits

Abstract

This paper is devoted to the semilinear stochastic wave equation

$$\begin{gathered} \square U = F(U) + H on {\mathbb{R}^{n + 1}}, \hfill \\ U|\left\{ {t < 0} \right\} = 0 \hfill \\ \end{gathered} $$

((0.1))

where F is smooth and globally Lipschitz, □ denotes the d’Alembertian $\partial _t^2 - \partial _{{x_1}}^2 - \ldots - \partial _{{x_n}}^2$, and H is a generalized stochastic process with support in the half space T = ℝⁿ × [0, ∞). That is, H is a weakly measurable map of some probability space (Ω, Σ, µ) with values in the space of Schwartz distributions D′(ℝⁿ⁺¹) which vanishes on {t < 0} almost surely. It is well known [27] that the solution to the linear wave equation

$$\begin{gathered} \square V = H on {\mathbb{R}^{n + 1}}, \hfill \\ V|\left\{ {t < 0} \right\} = 0 \hfill \\ \end{gathered} $$

((0.2))

is a generalized stochastic process in many cases, for example when the space dimension n ≥ 2, and H is a space-time white noise on T. Thus, when dealing with problem (0.1), we are faced with nonlinear operations on Schwartz distributions.