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2013 On the $L_q(L_p)$-regularity and Besov smoothness of stochastic parabolic equations on bounded Lipschitz domains
Petru Cioica, Kyeong-Hun Kim, Kijung Lee, Felix Lindner
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Electron. J. Probab. 18: 1-41 (2013). DOI: 10.1214/EJP.v18-2478


We investigate the regularity of linear stochastic parabolic equations with zero Dirichlet boundary condition on bounded Lipschitz domains $\mathcal{O}\subset \mathbb{R}^d$ with both theoretical and numerical purpose. We use N.V. Krylov’s framework of stochastic parabolic weighted Sobolev spaces $\mathfrak{H}^{\gamma,q}_{p,\theta} (\mathcal{O},T)$ The summability parameters $p$ and $q$ in space and time may differ. Existence and uniqueness of solutions in these spaces is established and the Hölder regularity in time is analysed. Moreover, we prove a general embedding of weighted $L_p(\mathcal{O})$ -Sobolev spaces into the scale of Besov spaces $B^\alpha_{\tau,\tau}(\mathcal{O})$ $1/\tau=\alpha/d+1/p$, $\alpha>0$. This leads to a Hölder-Besov regularity result for the solution process. The regularity in this Besov scale determines the order of convergence that can be achieved by certain nonlinear approximation schemes.


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Petru Cioica. Kyeong-Hun Kim. Kijung Lee. Felix Lindner. "On the $L_q(L_p)$-regularity and Besov smoothness of stochastic parabolic equations on bounded Lipschitz domains." Electron. J. Probab. 18 1 - 41, 2013.


Accepted: 13 September 2013; Published: 2013
First available in Project Euclid: 4 June 2016

zbMATH: 1309.60067
MathSciNet: MR3109621
Digital Object Identifier: 10.1214/EJP.v18-2478

Primary: 60H15
Secondary: 35R60 , 46E35

Keywords: $L_q(L_p)$-theory , adaptive numerical method , Besov space , Embedding theorem , H{\"o}lder regularity in time , Lipschitz domain , nonlinear approximation , quasi-Banach space , Stochastic partial differential equation , ‎wavelet , weighted Sobolev space

Vol.18 • 2013
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