发表机构
University of Ottawa（渥太华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究带非线性漂移和乘性高斯噪声的随机波动方程空间平均的高斯波动，建立定量中心极限定理，误差界分别为可积情形下的R^{-d/2}和Riesz核下的R^{-\eta/2}，并获泛函中心极限定理。
AI 中文摘要
本文研究了一维和二维空间中带非线性漂移和乘性高斯噪声的随机波动方程解的 spatial averages 的高斯波动。噪声在时间上是白的，其空间协方差要么是可积的，要么由 Riesz 核给出；一维情形也包含时空白噪声。我们建立了空间遍历性和重标度协方差的收敛性，并证明了半径为 $R$ 的球上中心化空间平均的定量中心极限定理。在可积情形下，全变差距离的界为 $R^{-d/2}$ 阶，对于阶为 $\eta$ 的 Riesz 核，界为 $R^{-\eta/2}$ 阶。我们还在连续函数空间中获得了泛函中心极限定理。我们的方法结合了二阶高斯 Poincaré 不等式和关于第二 Malliavin 导数的空间积分估计，利用了波核的紧支撑性。进一步基于 Clark-Ocone 公式的论证用于控制漂移项的贡献。
英文摘要
In this article, we study Gaussian fluctuations of spatial averages of the solution to the stochastic wave equation with a nonlinear drift and multiplicative Gaussian noise in dimensions one and two. The noise is white in time, and its spatial covariance is either integrable or given by a Riesz kernel; space-time white noise in dimension one is also included. We establish spatial ergodicity and convergence of the rescaled covariance, and prove a quantitative central limit theorem for the centered spatial average over a ball of radius $R$. The bounds in total variation distance are of order $R^{-d/2}$ in the integrable case and $R^{-β/2}$ for a Riesz kernel of order $β$. We also obtain a functional central limit theorem in the space of continuous functions. Our approach combines a second-order Gaussian Poincaré inequality with spatially integrated estimates for the second Malliavin derivative, exploiting the compact support of the wave kernel. Further arguments based on the Clark-Ocone formula are used to control the drift contributions.
Comments31 pages
