Regularity Theory of Stochastic Partial Differential Equations in (Quasi-)Banach Spaces
Final Report Abstract
Stochastic partial differential equations (SPDEs, for short) are mathematical models for evolutions in space and time, which are influenced by noise. They are aimed at describing phenomena in physics, chemistry, economics, and many other disciplines. Although we can prove existence and uniqueness of a solution to various classes of such equations, in general, we do not have an explicit representation of this solution. Thus, in order to make those models ready to use for applications, we need efficient numerical methods for approximating their solutions. The efficiency of approximation methods is closely related to the regularity of the target function. On the one hand, the Sobolev regularity governs the convergence rate of schemes based on uniform discretizations. On the other hand, the convergence rate of the best n-term approximation is determined by the regularity in special scales of Besov spaces, the so-called non-linear approximation scales. As a rule of thumb, if the regularity of the target function in these scales exceeds its corresponding Sobolev regularity, then there is room for improvement by switching from uniform to non-uniform, hence, adaptive strategies. Preliminary investigations have shown that solutions to second order SPDEs with zero Dirichlet boundary condition follow this pattern, and that, therefore, adaptivity can pay in the context of SPDEs. This project addressed some fundamental questions towards a theoretical underpinning of adaptive methods for SPDEs by extending and consolidating the corresponding regularity theory in Besov spaces from the non-linear approximation lines. The first main achievement points at the difference between the Besov regularity that was achievable for second order SPDEs on smooth domains and the relatively poor results for equations on general bounded Lipschitz domains. By using wavelet characterizations, up to a certain amount, the regularity analysis in Besov spaces can be traced back to regularity estimates in suitable weighted Sobolev spaces. The weights have to be designed in such a way that they capture the influence and interaction of the two main sources for spatial singularities: the incompatibility of noise and boundary condition and the bad influence of boundary singularities. For the first time the investigators manage to describe these effects for the solution of the stochastic heat equation, the prime example of a second order parabolic SPDE, on a domain that is smooth except at one point. The range of admissible weight parameters depends on the strength of the boundary singularity and is sharp. These results are a crucial step forward towards filling the long-standing gap between the almost fully-fledged regularity theory for second order SPDEs on smooth domains and the little that is known about such equations on non-smooth domains of high practical relevance, like polygons and polyhedra. The second main achievement is the extension of Itô’s stochastic integration theory to quasi- Banach spaces that admit a decoupling inequality. This is the cornerstone for a semigroup approach to SPDEs in the Besov spaces from the non-linear approximation scale. These spaces are mostly not Banach spaces, but quasi-Banach spaces, and as such, they fail to be locally convex. As a consequence an integral that obeys the usual properties of the Bochner integral cannot be established. However, as the investigators show, the stochastic Itô integral can be constructed along the lines of the stochastic integration theory in Banach spaces recently developed by van Neerven, Veraar, and Weis. Thus, this construction does not depend on the local convexity of the underlying space. If the space is separated by its dual (like it is the case for the Besov spaces of interest), then this integral can be seen as a stochastic version of Pettis’ integral. In addition, the convergence analysis of spatially adaptive methods for SPDEs has been started and the regularity of Navier-Stokes equations in the Besov spaces from the non-linear approximation scale has been analysed with similar techniques.
Publications
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An Lp-estimate for the stochastic heat equation on an angular domain in R2 , 2016
P.A. Cioica-Licht, K.-H. Kim, K. Lee, and F. Lindner
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On the convergence analysis of the inexact linearly implicit Euler scheme for a class of SPDEs, Potential Anal. 44 (2016), no. 3, 473–495
P.A. Cioica, S. Dahlke, N. Döhring, U. Friedrich, S. Kinzel, F. Lindner, T. Raasch, K. Ritter, and R.L. Schilling
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Besov regularity for the stationary Navier-Stokes equation on bounded Lipschitz domains, Applicable Analysis
F. Eckhardt, P.A. Cioica-Licht, S. Dahlke
