非线性随机波动方程系统空间平均的高斯波动

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

摘要: 在本文中，我们研究了由m维高斯时空白噪声驱动下的d个随机波动方程组成的系统。设该系统的解为u(t, x)=\left(u_1(t, x), \cdots, u_d(t, x)\right)。我们证明了当R趋于无穷时，空间平均向量 \left(R^-\frac12 \int_-R^R u_1(t, x) \mathrmd x, \cdots, R^-\frac12 \int_-R^R u_d(t, x) \mathrmd x\right) 在Wasserstein距离下收敛到一个高斯随机向量. 此外，我们还证明了与其对应的时间上的泛函中心极限定理.

Abstract: 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.