Poincar\'e inequality, and central limit theorems for parabolic stochastic partial differential equations
Poincar\'e inequality, and central limit theorems for parabolic stochastic partial differential equations
复制标题
DOI：
--
复制
发表时间：
2019-12
期刊：
影响因子：
--
通讯作者：
Le Chen;D. Khoshnevisan;D. Nualart;Fei Pu
中科院分区：
文献类型：
--
作者：
Le Chen;D. Khoshnevisan;D. Nualart;Fei Pu
设u(t，，x)tg0，x in{R}^d}表示一个由时间白噪声驱动的$d维非线性随机热方程解，它是一个有限的Borel测度，满足Dalang条件.我们证明了两个形式为$N^-d}\int_\mathbb{R}^d}g(u(t\，x))\psi(x/N)\mathm{d}x$as$N\right tarrow\inty$的占用域的泛函中心极限定理，其中$g$遍历L^2(R}^d)$中的Lipschitz函数类.该证明利用Poincar-e型不等式、Malliavin微积分、紧性变元和保罗·L对布朗运动的经典刻画作为唯一的平均零连续的L事件过程.我们的结果推广了Huang等人的中心极限定理，当$g(U)=u$和$\psi=\mathbf{1}{[0，1]^d}$时，中心极限定理在$g(U)=u$和$\psi=\mathbf{[0，1]^d}$时成立.
Let $\{u(t\,,x)\}_{t\ge 0, x\in \mathbb{R}^d}$ denote the solution of a $d$-dimensional nonlinear stochastic heat equation that is driven by a Gaussian noise, white in time with a homogeneous spatial covariance that is a finite Borel measure $f$ and satisfies Dalang's condition. We prove two general functional central limit theorems for occupation fields of the form $N^{-d} \int_{\mathbb{R}^d} g(u(t\,,x)) \psi(x/N)\, \mathrm{d} x$ as $N\rightarrow \infty$, where $g$ runs over the class of Lipschitz functions on $\mathbb{R}^d$ and $\psi\in L^2(\mathbb{R}^d)$. The proof uses Poincar\'e-type inequalities, Malliavin calculus, compactness arguments, and Paul L\'evy's classical characterization of Brownian motion as the only mean zero, continuous L\'evy process. Our result generalizes central limit theorems of Huang et al \cite{HuangNualartViitasaari2018,HuangNualartViitasaariZheng2019} valid when $g(u)=u$ and $\psi = \mathbf{1}_{[0,1]^d}$.