SIAM Journal on Mathematical Analysis, Volume 53, Issue 2, Page 2084-2133, January 2021.
Suppose that $\{u(t\,, x)\}_{t >0, x \in{\mathbb{R}}^d}$ is the solution to a $d$-dimensional parabolic Anderson model with delta initial condition and driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance given by a nonnegative-definite measure $f$ which satisfies Dalang's condition. Let ${\bf {{\it p}_t(x):= (2\pi t)^{-d/2}\exp\{-\|x\|^2/(2t)\}$ denote the standard Gaussian heat kernel on ${\mathbb{R}}^d$. We prove that for all $t>0$, the process $U(t):=\{u(t\,, x)/{\bf {{\it p}_t(x): x\in {\mathbb{R}}^d\}$ is stationary using the Feynman--Kac formula and is ergodic under the additional condition $\hat{f}\{0\}=0$, where $\hat{f}$ is the Fourier transform of $f$. Moreover, using the Malliavin--Stein method, we investigate various central limit theorems (CLTs) for $U(t)$ based on the quantitative analysis of $f$. In particular, when $f$ is given by the Riesz kernel, i.e., $f(\d x) = \|x\|^{-\beta}\d x$, we obtain a multiple phase transition for the CLT for $U(t)$ from $\beta\in(0\,,1)$ to $\beta=1$ to $\beta\in(1\,,d\wedge 2)$.

中文翻译：

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$ d \ geq 1 $维中具有Delta初始条件的抛物线Anderson模型的空间平稳性，遍历性和CLT

SIAM数学分析杂志，第53卷，第2期，第2084-2133页，2021年1月。
假设$ \ {u（t \ ,, x）\} _ {t> 0，x \ in {\ mathbb {R}} ^ d} $是具有增量的$ d $维抛物线安德森模型的解初始条件由高斯噪声驱动，该噪声在时间上为白色，并且具有满足大朗条件的非负定性度量$ f $给出的空间均匀协方差。令$ {\ bf {{\ it p} _t（x）：=（2 \ pi t）^ {-d / 2} \ exp \ {-\ | x \ | ^ 2 /（2t）\} $表示$ {\ mathbb {R}} ^ d $上的标准高斯热核。我们证明对于所有$ t> 0 $，过程$ U（t）：= \ {u（t \ ,, x）/ {\ bf {{\ it p} _t（x）：x \ in {\ mathbb {R}} ^ d \} $使用Feynman-Kac公式是固定的，并且在附加条件$ \ hat {f} \ {0 \} = 0 $下是遍历的，其中$ \ hat {f} $是$ f $的傅立叶变换。此外，使用Malliavin-Stein方法，我们基于对$ f $的定量分析，研究了$ U（t）$的各种中心极限定理（CLT）。特别是，当$ f $由Riesz内核给出时，即$ f（\ dx）= \ | x \ | ^ {-\ beta} \ dx $，我们为$ U获得了CLT的多相转换。 （t）$从$ \ beta \ in（0 \ ,, 1）$到$ \ beta = 1 $到$ \ beta \ in（1 \ ,, d \ wedge 2）$。