Consider the parabolic Anderson model of the form ${\mathrm{\partial }}_{t}u=\frac{1}{2}\mathrm{\Delta }u+u\eta$, where $u=u(t,x)$ for t>0 and $x\in {\mathbb{R}}^d$ with $u(0)={\delta }_0$, and η is a centered Gaussian noise that is white in time and has a spatially homogeneous covariance given by a nonnegative-definite measure f that satisfies Dalang's condition. Let ${\mathit{p}}_t(x):=(2\pi t{)}^{-d/2}\exp \{-\Vert x{\Vert }^2/(2t)\}$ denote the standard Gaussian heat kernel on ${\mathbb{R}}^d$ and set $U(t,x):=u(t,x)/{\mathit{p}}_t(x)$ for all t>0 and $x\in {\mathbb{R}}^d$. In this paper, we present an almost sure central limit theorem (ASCLT) and a functional ASCLT for spatial averages of the form ${\int }_{[0,N{]}^d}U(t,x)\mathrm{d}x$ as $N\to \mathrm{\infty }$ for fixed t>0 based on the quantitative analysis of f. In particular, when f is given by a Riesz kernel, that is, $f(\mathrm{d}x)=\Vert x{\Vert }^{-\beta }\mathrm{d}x$ for some $\beta \in (0,d\wedge 2)$, we can also obtain the ASCLT.

考虑形式的抛物线安德森模型${\mathrm{\partial }}_{吨}你=\frac{\mathrm{1个}}{\mathrm{2个}}\mathrm{△}你+你\eta$， 在哪里$你=你(吨,X)$对于t >0 和$X\in {\mathbb{R}}^d$和$你(0)={\delta }_0$，并且η是居中的高斯噪声，它在时间上是白色的，并且具有由满足 Dalang 条件的非负定测度f给出的空间齐次协方差。让${\mathit{p}}_{吨}(X):=(\mathrm{2个}\pi 吨{)}^{-d/\mathrm{2个}}\exp \{-\Vert X{\Vert }^{\mathrm{2个}}/(\mathrm{2个}吨)\}$表示标准高斯热核${\mathbb{R}}^d$并设置$ü(吨,X):=你(吨,X)/{\mathit{p}}_{吨}(X)$对于所有t >0 和$X\in {\mathbb{R}}^d$. 在本文中，我们提出了一个几乎确定的中心极限定理 (ASCLT) 和一个用于形式空间平均值的函数 ASCLT${\int }_{[0,否{]}^d}ü(吨,X)\mathrm{d}X$作为$否\to \mathrm{\infty }$对于基于f的定量分析的固定t >0 。特别地，当f由 Riesz 内核给出时，即$F(\mathrm{d}X)=\Vert X{\Vert }^{-\beta }\mathrm{d}X$对于一些$\beta \in (0,d\wedge \mathrm{2个})$，我们也可以获得ASCLT。