Delta初始条件下抛物型Anderson模型的空间遍历性和中心极限定理
Spatial ergodicity and central limit theorems for parabolic Anderson model with delta initial condition
摘要：
用$\{u(t\,, x)\}_{t >0, x \in\mathbb{R}}$表示$δ_0$为初始条件、$\mathbb{R}_+\times\mathbb{R}$上的时空白噪声驱动的抛物型Anderson模型的解，用$p_t(x):= (2πt)^{-1/2}\exp\{-x^2/(2t)\}$表示直线上的标准高斯热核.我们使用一个非平凡的适应我们的同伴论文\cite{CKNP,CKNP_b}的方法，以证明随机场$x\mapsto u(t\,,x)/p_t(x)$是遍历的每一个$t >0$。在Huang，Nualart，and Viitasaari \cite{HNV2018}中引入的Malliavin-Stein方法的基础上，我们建立了一个相关的定量中心极限定理.
英文摘要：
Let $\{u(t\,, x)\}_{t >0, x \in\mathbb{R}}$ denote the solution to the parabolic Anderson model with initial condition $δ_0$ and driven by space-time white noise on $\mathbb{R}_+\times\mathbb{R}$, and let $p_t(x):= (2πt)^{-1/2}\exp\{-x^2/(2t)\}$ denote the standard Gaussian heat kernel on the line. We use a non-trivial adaptation of the methods in our companion papers \cite{CKNP,CKNP_b} in order to prove that the random field $x\mapsto u(t\,,x)/p_t(x)$ is ergodic for every $t >0$. And we establish an associated quantitative central limit theorem following the approach based on the Malliavin-Stein method introduced in Huang, Nualart, and Viitasaari \cite{HNV2018}.
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