Gaussian Fluctuations of Spatial Averages for a Non-Linear System of Stochastic Wave Equations

In this paper, we study a system of d stochastic wave equations driven by m-dimensional space-time white noise. Let u(t, x)=\left(u_1(t, x), \cdots, u_d(t, x)\right) be the solution to the system. We show that the vector of spatial averages \left(R^-1 / 2 \int_-R^R u_1(t, x) \mathrmd x, \cdots, R^-1 / 2 \int_-R^R u_d(t, x) \mathrmd x\right) converges in the Wasserstein distance to a Gaussian random vector as R tends to infinity. And we also show an associated functional central limit theorem in time.