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
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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) = (u_1(t,x), \cdots, u_d(t,x)) . 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)=(u_1(t,x),\cdots,u_d(t,x)) 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) \textdx,\cdots,R^-1/2\int_-R^R u_d(t, x) \textdx\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.
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