27. chen.khoshnevisan.ea:21:spatial#
Central limit theorems for spatial averages of the stochastic heat equation via Malliavin-Stein’s method
Le Chen, Davar Khoshnevisan, David Nualart, and Fei Pu
Abstract 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
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.
MSC 2010 subject classifications: Primary. 60H15; Secondary. 60H07, 60F05.
Keywords: stochastic heat equation, ergodicity, central limit theorem, Malliavin calculus, Stein’s method.
[CKNP21b] Chen, Le, Khoshnevisan, Davar, Nualart, David & Pu, Fei (2023) ‘Central limit theorems for spatial averages of the stochastic heat equation via Malliavin-Stein’s method’, Stoch. Partial Differ. Equ. Anal. Comput. 11, 122–176. <https://doi.org/10.1007/s40072-021-00224-8>
@article{chen.khoshnevisan.ea:23:central,
author = {Chen, Le and Khoshnevisan, Davar and Nualart, David and Pu, Fei},
title = {Central limit theorems for spatial averages of the stochastic heat equation via {M}alliavin-{S}tein's method},
journal = {Stoch. Partial Differ. Equ. Anal. Comput.},
fjournal = {Stochastic Partial Differential Equations. Analysis and Computations},
volume = {11},
year = {2023},
number = {1},
pages = {122--176},
issn = {2194-0401},
mrclass = {60H15 (35K05 60F05 60H07)},
mrnumber = {4563698},
doi = {10.1007/s40072-021-00224-8},
url = {https://doi.org/10.1007/s40072-021-00224-8}
}
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