论文标题
与时间无关的噪声的中心限制定理用于热方程：规则和粗糙的情况
Central limit theorems for heat equation with time-independent noise: the regular and rough cases
论文作者
论文摘要
在本文中，我们调查了解决方案对抛物线模型的空间积分的渐近行为，随着时间噪声在尺寸$ d \ geq 1 $中，随着积分的域变大。我们考虑3例：（a）噪声具有可整合协方差函数的情况； （b）当Riesz内核给出噪声的协方差时； （c）粗糙噪声的情况，即用索引$ h \ in（\ frac {1} {4}，\ frac {1} {2}）$ in Dimension $ d = 1 $。在每种情况下，我们都会确定空间积分方差的数量级，我们通过将其总变化距离估计到标准正态分布，证明了归一化空间积分的定量中心极限定理，我们给出了相应的功能极限结果。
In this article, we investigate the asymptotic behaviour of the spatial integral of the solution to the parabolic Anderson model with time independent noise in dimension $d\geq 1$, as the domain of the integral becomes large. We consider 3 cases: (a) the case when the noise has an integrable covariance function; (b) the case when the covariance of the noise is given by the Riesz kernel; (c) the case of the rough noise, i.e. fractional noise with index $H \in (\frac{1}{4},\frac{1}{2})$ in dimension $d=1$. In each case, we identify the order of magnitude of the variance of the spatial integral, we prove a quantitative central limit theorem for the normalized spatial integral by estimating its total variation distance to a standard normal distribution, and we give the corresponding functional limit result.
