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BMO and Morrey–Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations / Jiang-lun, Wu

Journal of Differential Equations, Volume: 266, Issue: 5, Pages: 2666 - 2717

Swansea University Author: Jiang-lun, Wu

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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...

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Published in: Journal of Differential Equations
ISSN: 0022-0396
Published: Elsevier BV 2019
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URI: https://cronfa.swan.ac.uk/Record/cronfa39312
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spelling 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}&apos;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
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 2019-02-15T03:59:11Z
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