Central limit theorems for spatial averages of the stochastic heat equation via Malliavin-Stein's method
Central limit theorems for spatial averages of the stochastic heat equation via Malliavin-Stein's method
复制标题
DOI：
10.1007/s40072-021-00224-8
复制
发表时间：
2021-11-26
影响因子：
1.5
通讯作者：
Pu, Fei
中科院分区：
文献类型：
--
作者：
Chen, Le;Khoshnevisan, Davar;Pu, Fei
假设{u(t, x))（t>0，x是R中的一个元素）d是一个d维随机热方程的解，该方程由高斯噪声驱动，该高斯噪声在时间上是白色的，并且具有满足大朗条件的空间齐次协方差。本文的目的是利用Malliavin微积分与Stein的正态逼近方法相结合，建立N—(d)积分[0，N](d) g(u(t, x)) dx为N- >∞时，g是一个lipschitz -连续函数或属于一类局部lipschitz函数的空间平均的定量中心极限定理。我们的结果包括KPZ方程Hopf-Cole解的中心极限定理。我们还建立了这些空间平均值的泛函中心极限定理。
Suppose that {u(t, x))(t>0,x is an element of 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) integral[0,N](d) g(u(t, x)) dx, as N -> infinity, 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 Hopf-Cole solution to KPZ equation. We also establish a functional central limit theorem for these spatial averages.