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BMO and Morrey–Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations
Guangying Lv,
Hongjun Gao,
Jinlong Wei,
Jiang-lun Wu 
Journal of Differential Equations, Volume: 266, Issue: 5, Pages: 2666 - 2717
Swansea University Author:
Jiang-lun Wu
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DOI (Published version): 10.1016/j.jde.2018.08.042
Abstract
In this paper, we are aiming to prove several regularity results for the following stochastic fractional heat equations with additive noises \bessdu_t(x)=\Delta^{\frac{\alpha}{2}} u_t(x)dt+g(t,x)d\eta_t,\ \ \ u_0=0,\ \ \ t\in(0,T], \, x\in G, \eessfor a random field $u:(t,x)\in [0,T]\times G\mapsto...
| Published in: | Journal of Differential Equations |
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| ISSN: | 0022-0396 |
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Elsevier BV
2019
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<?xml version="1.0"?><rfc1807><datestamp>2020-07-27T15:19:49.9530742</datestamp><bib-version>v2</bib-version><id>39312</id><entry>2018-04-04</entry><title>BMO and Morrey–Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations</title><swanseaauthors><author><sid>dbd67e30d59b0f32592b15b5705af885</sid><ORCID>0000-0003-4568-7013</ORCID><firstname>Jiang-lun</firstname><surname>Wu</surname><name>Jiang-lun Wu</name><active>true</active><ethesisStudent>false</ethesisStudent></author></swanseaauthors><date>2018-04-04</date><deptcode>SMA</deptcode><abstract>In this paper, we are aiming to prove several regularity results for the following stochastic fractional heat equations with additive noises \bessdu_t(x)=\Delta^{\frac{\alpha}{2}} u_t(x)dt+g(t,x)d\eta_t,\ \ \ u_0=0,\ \ \ t\in(0,T], \, x\in G, \eessfor a random field $u:(t,x)\in [0,T]\times G\mapsto u(t,x)=:u_t(x)\in\mathbb{R}$, where $\Delta^{\frac{\alpha}{2}}:=-(-\Delta)^{\frac{\alpha}{2}}, \alpha\in(0,2]$, is the fractional Laplacian, $T\in(0,\infty)$ is arbitrarily fixed, $G\subset\mathbb{R}^d$ is a bounded domain,$g:[0,T]\times G\times\Omega\to\mathbb{R}$ is a joint measurable coefficient, and $\eta_t, t\in[0,\infty)$, is either a Brownian motion or a L\'evy process on a given filtered probability space $(\Omega,\mathcal{F},P;\{\mathcal{F}_t\}_{t\in[0,T]})$. To this end, we derive the BMO estimates and Morrey-Campanato estimates, respectively, for stochastic singular integral operators arising from the equations concerned. Then, by utilising the embedding theory between the Campanato space and the H\"older space, we establish the controllability of the norm of the space $C^{\theta,\theta/2}(\bar D)$, where $\theta\ge0,\bar D=[0,T]\times\bar G$. With all these in hand, we are able to show that the $q$-th order BMO quasi-norm of the $\frac{\alpha}{q_0}$-order derivative of the solution $u$ is controlled by the norm of $g$ under the condition that $\eta_t$ is a L\'evy process. Finally, we derive the Schauder estimate for the $p$-moments of the solution of the above stochastic fractional heat equations driven by L\'evy noise.</abstract><type>Journal Article</type><journal>Journal of Differential Equations</journal><volume>266</volume><journalNumber>5</journalNumber><paginationStart>2666</paginationStart><paginationEnd>2717</paginationEnd><publisher>Elsevier BV</publisher><issnPrint>0022-0396</issnPrint><keywords>Anomalous diffusion; It\^{o}&apos;s formula; BMO estimates; Morrey-Campanato estimates; Schauder estimate</keywords><publishedDay>15</publishedDay><publishedMonth>2</publishedMonth><publishedYear>2019</publishedYear><publishedDate>2019-02-15</publishedDate><doi>10.1016/j.jde.2018.08.042</doi><url/><notes/><college>COLLEGE NANME</college><department>Mathematics</department><CollegeCode>COLLEGE CODE</CollegeCode><DepartmentCode>SMA</DepartmentCode><institution>Swansea University</institution><apcterm/><lastEdited>2020-07-27T15:19:49.9530742</lastEdited><Created>2018-04-04T16:02:38.7566184</Created><authors><author><firstname>Guangying</firstname><surname>Lv</surname><order>1</order></author><author><firstname>Hongjun</firstname><surname>Gao</surname><order>2</order></author><author><firstname>Jinlong</firstname><surname>Wei</surname><order>3</order></author><author><firstname>Jiang-lun</firstname><surname>Wu</surname><orcid>0000-0003-4568-7013</orcid><order>4</order></author></authors><documents><document><filename>0039312-04042018160715.pdf</filename><originalFilename>LvGWWu.pdf</originalFilename><uploaded>2018-04-04T16:07:15.6000000</uploaded><type>Output</type><contentLength>444222</contentLength><contentType>application/pdf</contentType><version>Accepted Manuscript</version><cronfaStatus>true</cronfaStatus><embargoDate>2019-08-31T00:00:00.0000000</embargoDate><documentNotes>Released under the terms of a Creative Commons Attribution Non-Commercial No Derivatives License (CC-BY-NC-ND).</documentNotes><copyrightCorrect>true</copyrightCorrect><language>eng</language></document></documents><OutputDurs/></rfc1807> |
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2020-07-27T15:19:49.9530742 v2 39312 2018-04-04 BMO and Morrey–Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations dbd67e30d59b0f32592b15b5705af885 0000-0003-4568-7013 Jiang-lun Wu Jiang-lun Wu true false 2018-04-04 SMA In this paper, we are aiming to prove several regularity results for the following stochastic fractional heat equations with additive noises \bessdu_t(x)=\Delta^{\frac{\alpha}{2}} u_t(x)dt+g(t,x)d\eta_t,\ \ \ u_0=0,\ \ \ t\in(0,T], \, x\in G, \eessfor a random field $u:(t,x)\in [0,T]\times G\mapsto u(t,x)=:u_t(x)\in\mathbb{R}$, where $\Delta^{\frac{\alpha}{2}}:=-(-\Delta)^{\frac{\alpha}{2}}, \alpha\in(0,2]$, is the fractional Laplacian, $T\in(0,\infty)$ is arbitrarily fixed, $G\subset\mathbb{R}^d$ is a bounded domain,$g:[0,T]\times G\times\Omega\to\mathbb{R}$ is a joint measurable coefficient, and $\eta_t, t\in[0,\infty)$, is either a Brownian motion or a L\'evy process on a given filtered probability space $(\Omega,\mathcal{F},P;\{\mathcal{F}_t\}_{t\in[0,T]})$. To this end, we derive the BMO estimates and Morrey-Campanato estimates, respectively, for stochastic singular integral operators arising from the equations concerned. Then, by utilising the embedding theory between the Campanato space and the H\"older space, we establish the controllability of the norm of the space $C^{\theta,\theta/2}(\bar D)$, where $\theta\ge0,\bar D=[0,T]\times\bar G$. With all these in hand, we are able to show that the $q$-th order BMO quasi-norm of the $\frac{\alpha}{q_0}$-order derivative of the solution $u$ is controlled by the norm of $g$ under the condition that $\eta_t$ is a L\'evy process. Finally, we derive the Schauder estimate for the $p$-moments of the solution of the above stochastic fractional heat equations driven by L\'evy noise. Journal Article Journal of Differential Equations 266 5 2666 2717 Elsevier BV 0022-0396 Anomalous diffusion; It\^{o}'s formula; BMO estimates; Morrey-Campanato estimates; Schauder estimate 15 2 2019 2019-02-15 10.1016/j.jde.2018.08.042 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University 2020-07-27T15:19:49.9530742 2018-04-04T16:02:38.7566184 Guangying Lv 1 Hongjun Gao 2 Jinlong Wei 3 Jiang-lun Wu 0000-0003-4568-7013 4 0039312-04042018160715.pdf LvGWWu.pdf 2018-04-04T16:07:15.6000000 Output 444222 application/pdf Accepted Manuscript true 2019-08-31T00:00:00.0000000 Released under the terms of a Creative Commons Attribution Non-Commercial No Derivatives License (CC-BY-NC-ND). true eng |
| title |
BMO and Morrey–Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations |
| spellingShingle |
BMO and Morrey–Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations Jiang-lun Wu |
| title_short |
BMO and Morrey–Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations |
| title_full |
BMO and Morrey–Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations |
| title_fullStr |
BMO and Morrey–Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations |
| title_full_unstemmed |
BMO and Morrey–Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations |
| title_sort |
BMO and Morrey–Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations |
| author_id_str_mv |
dbd67e30d59b0f32592b15b5705af885 |
| author_id_fullname_str_mv |
dbd67e30d59b0f32592b15b5705af885_***_Jiang-lun Wu |
| author |
Jiang-lun Wu |
| author2 |
Guangying Lv Hongjun Gao Jinlong Wei Jiang-lun Wu |
| format |
Journal article |
| container_title |
Journal of Differential Equations |
| container_volume |
266 |
| container_issue |
5 |
| container_start_page |
2666 |
| publishDate |
2019 |
| institution |
Swansea University |
| issn |
0022-0396 |
| doi_str_mv |
10.1016/j.jde.2018.08.042 |
| publisher |
Elsevier BV |
| document_store_str |
1 |
| active_str |
0 |
| description |
In this paper, we are aiming to prove several regularity results for the following stochastic fractional heat equations with additive noises \bessdu_t(x)=\Delta^{\frac{\alpha}{2}} u_t(x)dt+g(t,x)d\eta_t,\ \ \ u_0=0,\ \ \ t\in(0,T], \, x\in G, \eessfor a random field $u:(t,x)\in [0,T]\times G\mapsto u(t,x)=:u_t(x)\in\mathbb{R}$, where $\Delta^{\frac{\alpha}{2}}:=-(-\Delta)^{\frac{\alpha}{2}}, \alpha\in(0,2]$, is the fractional Laplacian, $T\in(0,\infty)$ is arbitrarily fixed, $G\subset\mathbb{R}^d$ is a bounded domain,$g:[0,T]\times G\times\Omega\to\mathbb{R}$ is a joint measurable coefficient, and $\eta_t, t\in[0,\infty)$, is either a Brownian motion or a L\'evy process on a given filtered probability space $(\Omega,\mathcal{F},P;\{\mathcal{F}_t\}_{t\in[0,T]})$. To this end, we derive the BMO estimates and Morrey-Campanato estimates, respectively, for stochastic singular integral operators arising from the equations concerned. Then, by utilising the embedding theory between the Campanato space and the H\"older space, we establish the controllability of the norm of the space $C^{\theta,\theta/2}(\bar D)$, where $\theta\ge0,\bar D=[0,T]\times\bar G$. With all these in hand, we are able to show that the $q$-th order BMO quasi-norm of the $\frac{\alpha}{q_0}$-order derivative of the solution $u$ is controlled by the norm of $g$ under the condition that $\eta_t$ is a L\'evy process. Finally, we derive the Schauder estimate for the $p$-moments of the solution of the above stochastic fractional heat equations driven by L\'evy noise. |
| published_date |
2019-02-15T03:49:54Z |
| _version_ |
1763752431955476480 |
| score |
11.035634 |