论文标题
通过Malliavin-Stein的方法，用于随机热方程的空间平均值的中心限制定理
Central limit theorems for spatial averages of the stochastic heat equation via Malliavin-Stein's method
论文作者
论文摘要
假设$ \ {u（t \ ,, x）_ {t> 0，x \ in \ mathbb {r}^d} $是解决$ d $维的随机热方程的解决方案，由高斯噪声驱动，高斯噪声在时间上是白色的，并且在空间上具有同质性的协调dalang的条件。 The purpose of this paper is to establish quantitative central limit theorems for spatial averages of the form $N^{-d} \int_{[0,N]^d} g(u(t\,,x))\, \mathrm{d} x$, as $N\rightarrow\infty$, where $g$ is a Lipschitz-continuous function or belongs to a class of局部 - lipschitz的函数，使用Malliavin conculus和Stein的方法的组合进行正常近似。我们的结果包括{\ it hopf-cole}解决方程的中心限制定理。我们还为这些空间平均值建立了功能性中心极限定理。
Suppose that $\{u(t\,, x)\}_{t >0, x \in\mathbb{R}^d}$ is the solution to a $d$-dimensional stochastic heat equation driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance that satisfies Dalang's condition. The purpose of this paper is to establish quantitative central limit theorems for spatial averages of the form $N^{-d} \int_{[0,N]^d} g(u(t\,,x))\, \mathrm{d} x$, as $N\rightarrow\infty$, where $g$ is a Lipschitz-continuous function or belongs to a class of locally-Lipschitz functions, using a combination of the Malliavin calculus and Stein's method for normal approximations. Our results include a central limit theorem for the {\it Hopf-Cole} solution to KPZ equation. We also establish a functional central limit theorem for these spatial averages.
