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

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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.