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Article

Asymptotic Distributions for Power Variations of the Solutions to Linearized Kuramoto–Sivashinsky SPDEs in One-to-Three Dimensions

1
School of Economics, Hangzhou Dianzi University, Hangzhou 310018, China
2
Zhiyuan College, Shanghai Jiao Tong University, Shanghai 200240, China
*
Author to whom correspondence should be addressed.
Symmetry 2021, 13(1), 73; https://doi.org/10.3390/sym13010073
Received: 22 November 2020 / Revised: 28 December 2020 / Accepted: 29 December 2020 / Published: 3 January 2021
We study the realized power variations for the fourth order linearized Kuramoto–Sivashinsky (LKS) SPDEs and their gradient, driven by the space–time white noise in one-to-three dimensional spaces, in time, have infinite quadratic variation and dimension-dependent Gaussian asymptotic distributions. This class was introduced-with Brownian-time-type kernel formulations by Allouba in a series of articles starting in 2006. He proved the existence, uniqueness, and sharp spatio-temporal Hölder regularity for the above class of equations in d=1,2,3. We use the relationship between LKS-SPDEs and the Houdré–Villaa bifractional Brownian motion (BBM), yielding temporal central limit theorems for LKS-SPDEs and their gradient. We use the underlying explicit kernels and spectral/harmonic analysis to prove our results. On one hand, this work builds on the recent works on the delicate analysis of variations of general Gaussian processes and stochastic heat equation driven by the space–time white noise. On the other hand, it builds on and complements Allouba’s earlier works on the LKS-SPDEs and their gradient. View Full-Text
Keywords: quadratic variation; power variation; linearized Kuramoto–Sivashinsky SPDEs; space–time white noise; weak convergence quadratic variation; power variation; linearized Kuramoto–Sivashinsky SPDEs; space–time white noise; weak convergence
MDPI and ACS Style

Wang, W.; Wang, D. Asymptotic Distributions for Power Variations of the Solutions to Linearized Kuramoto–Sivashinsky SPDEs in One-to-Three Dimensions. Symmetry 2021, 13, 73. https://doi.org/10.3390/sym13010073

AMA Style

Wang W, Wang D. Asymptotic Distributions for Power Variations of the Solutions to Linearized Kuramoto–Sivashinsky SPDEs in One-to-Three Dimensions. Symmetry. 2021; 13(1):73. https://doi.org/10.3390/sym13010073

Chicago/Turabian Style

Wang, Wensheng, and Dazhong Wang. 2021. "Asymptotic Distributions for Power Variations of the Solutions to Linearized Kuramoto–Sivashinsky SPDEs in One-to-Three Dimensions" Symmetry 13, no. 1: 73. https://doi.org/10.3390/sym13010073

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