Besov Regularity of Stochastic Partial Differential Equations on Bounded Lipschitz Domains
Petru Cioica
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Petru Cioica, Besov Regularity of Stochastic Partial Differential Equations on Bounded Lipschitz Domains (2015), Logos Verlag, Berlin, ISBN: 9783832591335
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Beschreibung / Abstract
Stochastic partial differential equations (SPDEs, for short) are the mathematical models of choice for space time evolutions corrupted by noise, as they frequently appear, for instance, in chemistry, physics and also in finance. Although in many settings it is known that the resulting SPDEs have a unique solution, in general, this solution is not given explicitly. Thus, in order to make those mathematical models ready to use for real life applications, suitable numerical algorithms are needed. Since realistic problems lead to systems involving thousands or even millions of unknowns, it would be tempting to use adaptive schemes based, e.g., on wavelets in order to increase efficiency. However, it is not a priori clear whether such adaptive strategies can outperform well-established uniform alternatives. Their theoretical justification requires a rigorous regularity analysis in so-called non-linear approximation scales of Besov spaces.
In this thesis the regularity of (semi-)linear second order SPDEs of It ^o type on general bounded Lipschitz domains is analysed. The non-linear approximation scales of Besov spaces are used to measure the regularity with respect to the space variable. In particular, it is shown that in specific situations the spatial Besov regularity in the non-linear approximation scales is generically higher than its corresponding classical Sobolev regularity. This indicates that it is worth developing spatially adaptive wavelet methods for solving SPDEs instead of using uniform alternatives. The time regularity is analysed as well. Results concerning the Hölder-Besov time-space regularity of the solution to the stochastic heat equation with additive noise on general bounded Lipschitz domains are derived. To this end, an Lq(Lp)-theory is established by a combination of results from the semigroup approach with techniques from the analytic approach. Moreover, a general embedding of weighted Sobolev spaces into Besov spaces from the non-linear approximation scales is established by means of wavelet techinques.
In this thesis the regularity of (semi-)linear second order SPDEs of It ^o type on general bounded Lipschitz domains is analysed. The non-linear approximation scales of Besov spaces are used to measure the regularity with respect to the space variable. In particular, it is shown that in specific situations the spatial Besov regularity in the non-linear approximation scales is generically higher than its corresponding classical Sobolev regularity. This indicates that it is worth developing spatially adaptive wavelet methods for solving SPDEs instead of using uniform alternatives. The time regularity is analysed as well. Results concerning the Hölder-Besov time-space regularity of the solution to the stochastic heat equation with additive noise on general bounded Lipschitz domains are derived. To this end, an Lq(Lp)-theory is established by a combination of results from the semigroup approach with techniques from the analytic approach. Moreover, a general embedding of weighted Sobolev spaces into Besov spaces from the non-linear approximation scales is established by means of wavelet techinques.
Inhaltsverzeichnis
- BEGINN
- Acknowledgement
- Introduction
- Motivation
- Overview of the relevant regularity theory
- The thesis in a nutshell
- Outline
- Preliminaries
- Some conventions
- Stochastic integration in UMD Banach spaces
- Function spaces
- Semigroups of linear operators
- Starting point: Linear SPDEs in weighted Sobolev spaces
- Stochastic parabolic weighted Sobolev spaces H,qp,(G,T)
- An Lp-theory of linear SPDEs on bounded Lipschitz domains
- Embeddings of weighted Sobolev spaces into Besov spaces
- Weighted Sobolev spaces and Sobolev spaces without weights
- Wavelet decomposition of Besov spaces on Rd
- Weighted Sobolev spaces and the non-linear approximation scale
- An alternative proof of Theorem 4.7
- Spatial Besov regularity of SPDEs on bounded Lipschitz domains
- Linear equations
- Semi-linear equations
- Space time regularity of the inhomogeneous heat equation with additive noise
- Space time regularity of elements from H,qp,(O,T)
- The spaces H,qp,(O,T) and SPDEs
- The stochastic heat equation in H,qp,(O,T)
- Zusammenfassung
- Notation
- Bibliography
- Index