论文标题
与时间无关的噪声的杂音安德森模型解的空间积分
Spatial integral of the solution to hyperbolic Anderson model with time-independent noise
论文作者
论文摘要
在本文中，我们研究了dimension $ d \ leq 2 $的解决方案的空间积分的渐近行为，因为该积分的域变得很大（固定时间$ t $）。该方程是由空间均匀的高斯噪声驱动的，高斯噪声的协方差函数要么可以集成，要么由Riesz内核给出。新颖的是噪声不取决于时间，这意味着无法使用ITô的Martingale理论。通过将Malliavin微积分与Stein的方法组合在一起，我们表明，通过适当的归一化和核心，解决方案的空间积分收敛到标准正态分布，通过估计总变化距离中这种收敛的速度。我们还证明了空间积分过程的相应功能极限定理。
In this article, we study the asymptotic behavior of the spatial integral of the solution to the hyperbolic Anderson model in dimension $d\leq 2$, as the domain of the integral gets large (for fixed time $t$). This equation is driven by a spatially homogeneous Gaussian noise, whose covariance function is either integrable, or is given by the Riesz kernel. The novelty is that the noise does not depend on time, which means that Itô's martingale theory for stochastic integration cannot be used. Using a combination of Malliavin calculus with Stein's method, we show that with proper normalization and centering, the spatial integral of the solution converges to a standard normal distribution, by estimating the speed of this convergence in the total variation distance. We also prove the corresponding functional limit theorem for the spatial integral process.
