Asymptotic Distributions for Power Variations of the Solutions to Linearized Kuramoto–Sivashinsky SPDEs in One-to-Three Dimensions
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. T...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-01-01
|
| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/13/1/73 |
| _version_ | 1797542522267893760 |
|---|---|
| author | Wensheng Wang Dazhong Wang |
| author_facet | Wensheng Wang Dazhong Wang |
| author_sort | Wensheng Wang |
| collection | DOAJ |
| description | 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 <inline-formula><math display="inline"><semantics><mrow><mi>d</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></semantics></math></inline-formula>. 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. |
| first_indexed | 2024-03-10T13:31:45Z |
| format | Article |
| id | doaj.art-311cc6325c884b88b9e26a3eaf934db7 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T13:31:45Z |
| publishDate | 2021-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-311cc6325c884b88b9e26a3eaf934db72023-11-21T07:55:31ZengMDPI AGSymmetry2073-89942021-01-011317310.3390/sym13010073Asymptotic Distributions for Power Variations of the Solutions to Linearized Kuramoto–Sivashinsky SPDEs in One-to-Three DimensionsWensheng Wang0Dazhong Wang1School of Economics, Hangzhou Dianzi University, Hangzhou 310018, ChinaZhiyuan College, Shanghai Jiao Tong University, Shanghai 200240, ChinaWe 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 <inline-formula><math display="inline"><semantics><mrow><mi>d</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></semantics></math></inline-formula>. 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.https://www.mdpi.com/2073-8994/13/1/73quadratic variationpower variationlinearized Kuramoto–Sivashinsky SPDEsspace–time white noiseweak convergence |
| spellingShingle | Wensheng Wang Dazhong Wang Asymptotic Distributions for Power Variations of the Solutions to Linearized Kuramoto–Sivashinsky SPDEs in One-to-Three Dimensions Symmetry quadratic variation power variation linearized Kuramoto–Sivashinsky SPDEs space–time white noise weak convergence |
| title | Asymptotic Distributions for Power Variations of the Solutions to Linearized Kuramoto–Sivashinsky SPDEs in One-to-Three Dimensions |
| title_full | Asymptotic Distributions for Power Variations of the Solutions to Linearized Kuramoto–Sivashinsky SPDEs in One-to-Three Dimensions |
| title_fullStr | Asymptotic Distributions for Power Variations of the Solutions to Linearized Kuramoto–Sivashinsky SPDEs in One-to-Three Dimensions |
| title_full_unstemmed | Asymptotic Distributions for Power Variations of the Solutions to Linearized Kuramoto–Sivashinsky SPDEs in One-to-Three Dimensions |
| title_short | Asymptotic Distributions for Power Variations of the Solutions to Linearized Kuramoto–Sivashinsky SPDEs in One-to-Three Dimensions |
| title_sort | asymptotic distributions for power variations of the solutions to linearized kuramoto sivashinsky spdes in one to three dimensions |
| topic | quadratic variation power variation linearized Kuramoto–Sivashinsky SPDEs space–time white noise weak convergence |
| url | https://www.mdpi.com/2073-8994/13/1/73 |
| work_keys_str_mv | AT wenshengwang asymptoticdistributionsforpowervariationsofthesolutionstolinearizedkuramotosivashinskyspdesinonetothreedimensions AT dazhongwang asymptoticdistributionsforpowervariationsofthesolutionstolinearizedkuramotosivashinskyspdesinonetothreedimensions |