BMO and Morrey-Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:266

Article

Lv, Guangying^1,2  Gao, Hongjun²  Wei, Jinlong³  Wu, Jiang-Lun⁴ 

[1] Henan Univ, Inst Appl Math, Kaifeng 475001, Henan, Peoples R China

[2] Nanjing Normal Univ, Sch Math Sci, Inst Math, Nanjing 210023, Jiangsu, Peoples R China

[3] Zhongnan Univ Econ & Law, Sch Stat & Math, Wuhan 430073, Hubei, Peoples R China

[4] Swansea Univ, Dept Math, Swansea SA2 8PP, W Glam, Wales

关键词: Anomalous diffusion;    Ito's formula;    BMO estimates;    Money-Campanato estimates;    Schauder estimate;   

DOI  :  10.1016/j.jde.2018.08.042

来源: Elsevier

【 摘 要 】

In this paper, we are aiming to prove several regularity results for the following stochastic fractional heat equations with additive noises du(t)(x)=Delta alpha/2 u(t)(x)dt + g(t,x)d eta(t), u(0)=0, t is an element of (0,T], x is an element of G, for a random field u : (t, x) is an element of [0, T] x G bar right arrow u(t , x) =: u(t)(x) is an element of R, where Delta alpha/2 -(-Delta) , alpha is an element of (0, 2], is the fractional Laplacian, T is an element of (0, infinity) is arbitrarily fixed, G subset of R-d is a bounded domain, g : [0, T] x G x Omega -> R is a joint measurable coefficient, and eta(t), t is an element of [0, infinity), is either a Brownian motion or a Levy process on a given filtered probability space (Omega,F P;{F-t}(t is an element of[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 utilizing the embedding theory between the Campanato space and the Holder space, we establish the controllability of the norm of the space C-theta,C-theta/2((D) over bar), where theta >= 0, (D) over bar = [0, T] x (G) over bar. With all these in hand, we are able to show that the q-th order BMO quasi-norm of the alpha/q0 -order derivative of the solution u is controlled by the norm of g under the condition that eta(t) is a Levy process. Finally, we derive the Schauder estimate for the p-moments of the solution of the above stochastic fractional heat equations driven by Levy noise. (C) 2018 Elsevier Inc. All rights reserved.

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