{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:42:07Z","timestamp":1760233327581,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,3]],"date-time":"2021-01-03T00:00:00Z","timestamp":1609632000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We study the realized power variations for the fourth order linearized Kuramoto\u2013Sivashinsky (LKS) SPDEs and their gradient, driven by the space\u2013time 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 H\u00f6lder regularity for the above class of equations in d=1,2,3. We use the relationship between LKS-SPDEs and the Houdr\u00e9\u2013Villaa 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\u2013time white noise. On the other hand, it builds on and complements Allouba\u2019s earlier works on the LKS-SPDEs and their gradient.<\/jats:p>","DOI":"10.3390\/sym13010073","type":"journal-article","created":{"date-parts":[[2021,1,3]],"date-time":"2021-01-03T19:54:46Z","timestamp":1609703686000},"page":"73","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Asymptotic Distributions for Power Variations of the Solutions to Linearized Kuramoto\u2013Sivashinsky SPDEs in One-to-Three Dimensions"],"prefix":"10.3390","volume":"13","author":[{"given":"Wensheng","family":"Wang","sequence":"first","affiliation":[{"name":"School of Economics, Hangzhou Dianzi University, Hangzhou 310018, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dazhong","family":"Wang","sequence":"additional","affiliation":[{"name":"Zhiyuan College, Shanghai Jiao Tong University, Shanghai 200240, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"6851","DOI":"10.1016\/j.jde.2015.08.033","article-title":"L-Kuramoto\u2013Sivashinsky SPDEs in one-to-three dimensions: L-KS kernel, sharp H\u00f6lder regularity, and Swift-Hohenberg law equivalence","volume":"259","author":"Allouba","year":"2015","journal-title":"J. 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