Regularity Theory of Stochastic Partial Differential Equations in (Quasi-)Banach Spaces

本文研究有界Lipschitz域上随机偏微分方程（SPDEs）的正则性估计。我们使用特定的（拟）Banach空间来测量解的光滑性。我们的分析是由在spde数值处理的背景下出现的一些基本问题所激发的。我们将我们的研究分为三个相互密切相关的部分。在前两部分中，我们使用特定的Besov空间尺度来度量解过程的空间规律性。这个尺度决定了自适应（小波）方案和其他非线性近似方法可以实现的收敛顺序。它主要由可和性参数小于1的Besov空间组成，因此是拟banach空间。在第一部分中，我们想要在加权Sobolev空间中推导出精细的正则性结果，通过嵌入策略得到期望的Besov正则性结果。在第二部分中，我们力求采用一种更直接的方法。为此，我们必须将众所周知的UMD-Banach空间的随机积分理论尽可能推广到拟banach空间。在项目的第三部分，我们希望推导加权Sobolev空间张量积的正则性估计，这将证明各向异性全时空自适应张量小波方法的使用是合理的。

This project is concerned with regularity estimates for stochastic partial differential equations (SPDEs, for short) on bounded Lipschitz domains. We use specific (quasi-)Banach spaces to measure the smoothness of the solution. Our analysis is motivated by some fundamental problems arising in the context of the numerical treatment of SPDEs We divide our investigations into three parts which are closely related to each other. In the first two parts we use a specific scale of Besov spaces to measure the spatial regularity of the solution process. This scale determines the convergence order that can be achieved by adaptive (wavelet) schemes and other non-linear approximation methods. It consists mostly of Besov spaces with summability parameter less than one, thus, of quasi-Banach spaces. In the first part we want to derive refined regularity results in weighted Sobolev spaces which yield the desired Besov regularity results by embedding strategies. In the second part, we strive for a more direct approach. To this end, the well-known theory of stochastic integration in UMD-Banach spaces has to be generalized as far as possible to quasi-Banach spaces. In the third part of the project, we want to derive regularity estimates in tensor products of weighted Sobolev spaces which would justify the use of anisotropic full space-time adaptive tensor wavelet methods.