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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

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 2019
Online Access: Check full text

URI: https://cronfa.swan.ac.uk/Record/cronfa39312
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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 $u:(t,x)\in [0,T]\times G\mapsto u(t,x)=:u_t(x)\in\mathbb{R}$, 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.
Keywords: Anomalous diffusion; It\^{o}'s formula; BMO estimates; Morrey-Campanato estimates; Schauder estimate.
Issue: 5
Start Page: 2666
End Page: 2717