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

本项目主要研究有界Lipschitz域上随机偏微分方程的正则性估计。我们使用特定的（拟）Banach空间来衡量的光滑性的解决方案。我们的分析是出于一些基本问题的背景下产生的数值处理的SPDE我们分为三个部分，这是密切相关的彼此的调查。在前两部分中，我们使用一个特定尺度的Besov空间来度量解的空间正则性。该尺度决定了自适应（小波）方案和其他非线性近似方法可以实现的收敛阶数。它主要由Besov空间与求和参数小于一个，因此，准Banach空间。在第一部分中，我们希望得到加权Sobolev空间中的精化正则性结果，通过嵌入策略得到所需的Besov正则性结果。在第二部分中，我们努力采取更直接的方法。为此，著名的随机积分理论在UMD-Banach空间已尽可能推广到准Banach空间。在项目的第三部分中，我们希望得到加权Sobolev空间的张量积的正则性估计，这将证明使用各向异性全时空自适应张量小波方法是合理的。

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.