Abstract
Fix \(d\in \{1,2\}\), we consider a d-dimensional stochastic wave equation driven by a Gaussian noise, which is temporally white and colored in space such that the spatial correlation function is integrable and satisfies Dalang’s condition. In this setting, we provide quantitative central limit theorems for the spatial average of the solution over a Euclidean ball, as the radius of the ball diverges to infinity. We also establish functional central limit theorems. A fundamental ingredient in our analysis is the pointwise \(L^p\)-estimate for the Malliavin derivative of the solution, which is of independent interest. This paper is another addendum to the recent research line of averaging stochastic partial differential equations.
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Notes
Note that \(\sigma _R\) depends on the parameter t and the conclusion “\(\sigma _R>0\) for each \(R>0\)” is ensured by condition (C3). The proof of this part is omitted here and can be done by following the same arguments as in [8, Lemma 3.4].
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Acknowledgements
D. Nualart is supported by NSF Grant DMS 1811181. The authors are thankful to two anonymous referees for the comments and suggestions. The authors are also very grateful to Raluca M. Balan for sending a long list of comments that have improved the paper.
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Nualart, D., Zheng, G. Central limit theorems for stochastic wave equations in dimensions one and two. Stoch PDE: Anal Comp 10, 392–418 (2022). https://doi.org/10.1007/s40072-021-00209-7
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DOI: https://doi.org/10.1007/s40072-021-00209-7