发表机构
Center for Applied Mathematics, Tianjin University; Department of Mathematics, Southern University of Science and Technology（天津大学应用数学中心; 南方科技大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对热核协方差高斯噪声驱动的随机热方程，证明了空间积分的定量和函数中央极限定理，速率为$R^{-d/2}$。
AI 中文摘要
本文研究了在$\mathbb R^d$（$d<4$）上，由具有非可分离协方差核$p_{|t-s|}(x-y)$的乘性高斯噪声驱动的随机热方程的Skorohod解的空间波动。我们证明了该解在任意固定时刻是严格平稳且空间遍历的。对于中心化空间积分$F_R(t)=\int_{\{|x|<R\}}(u(t,x)-1)dx$，我们证明了当$R\to\infty$时，$\mathbb E[F_R(t)F_R(s)]\sim K(t,s)R^d$。利用前两个Malliavin导数的矩估计和二阶高斯Poincaré不等式，我们建立了总变差距离下以速率$R^{-d/2}$收敛的定量中央极限定理。我们还证明了过程$\{R^{-d/2}F_R(t)\}_{t\geq0}$的函数中央极限定理。
英文摘要
In this article, we study the spatial fluctuations of the Skorohod solution to a stochastic heat equation on $\mathbb R^d$ for $d<4$, driven by multiplicative Gaussian noise with the non-separable covariance kernel $p_{|t-s|}(x-y)$. We prove that the solution is strictly stationary and spatially ergodic at every fixed time. For the centered spatial integral $F_R(t)=\int_{\{|x|<R\}}(u(t,x)-1)dx,$ we show that $\mathbb E[F_R(t)F_R(s)]\sim K(t,s)R^d$ as $R\to\infty$. Using moment estimates for the first two Malliavin derivatives and a second-order Gaussian Poincaré inequality, we establish a quantitative central limit theorem in total variation distance with rate $R^{-d/2}$. We also prove a functional central limit theorem for the process $\{R^{-d/2}F_R(t)\}_{t\geq0}$.
