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Approximation and regularity of stochastic PDEs. (English) Zbl 1260.60003

Berichte aus der Mathematik. Aachen: Shaker Verlag; Dresden: Univ. Dresden, Fakultät Mathematik und Naturwissenschaften (Diss.) (ISBN 978-3-8440-0172-3/pbk). ii, 141 p. (2011).
Summary: This thesis treats specific questions concerning the approximation and the regularity of the solutions to linear parabolic stochastic partial differential equations (SPDEs). The thesis consists of two main parts.
The first main part (Chapter 2) is concerned with the so-called weak order of convergence of a uniform space-time discretization scheme for the stochastic heat equation with zero Dirichlet boundary condition on a convex polygonal domain \({\mathcal O}\subset\mathbb{R}^d\) driven by an infinite-dimensional Lévy process. A discretization \((X^n_h)_{n\in\{0,1,\dots, N\}}\) of the \(L_2({\mathcal O})\)-valued solution process \((X_t)_{t\in [0,T]}\) is defined by the finite element method in space (parameter \(h>0\)) and an implicit Euler scheme in time (parameter \(\Delta t= T/N\)). For suitable test functions \(\varphi\) defined on \(L_2({\mathcal O})\) we derive an integral representation for the weak error \(|\mathbb{E}\varphi(X^N_h)- \mathbb{E}\varphi(X_t)|\) and prove that it decays with order \(O(h^{2\gamma}+ (\Delta t)^\gamma)\) for certain \(\gamma\) depending on the space dimension \(d\) and on the spatial regularity of the driving Lévy process.
The investigations in the second main part (Chapters 3 and 4) are motivated by the question whether adaptive and other nonlinear approximation methods for SPDEs pay off in the sense that they admit better convergence rates than uniform approximation methods. In Chapter 3, we consider linear parabolic SPDEs on general bounded Lipschitz domains \({\mathcal O}\subset\mathbb{R}^d\) driven by infinite-dimensional Wiener processes. We prove a result on the spatial regularity of the solution within the scale of Besov spaces \(B^\alpha_{\tau,\tau}({\mathcal O})\), \(\alpha> 0\), \({1\over\tau}= {\alpha\over d}+{1\over p}\), \(p\geq 2\) fixed. This scale is closely connected to the order of convergence in \(L_p({\mathcal O})\) of certain nonlinear approximation methods. The proof of the result is based on a combination of weighted Sobolev estimates and characterizations of Besov spaces by wavelet expansions. In Chapter 4, we consider the stochastic heat equation with zero Dirichlet boundary condition on a (possibly non-convex) polygonal domain \({\mathcal O}\subset\mathbb{R}^2\) driven by an infinite-dimensional Wiener process. Based on a classical result by P. Grisvard and a Laplace transform argument, we prove that the solution process can be decomposed into a regular part with maximal spatial \(L_2\)-Sobolev regularity and a singular part whose spatial \(L_2\)-Sobolev regularity is restricted due to the shape of the domain \({\mathcal O}\). This leads to an explicit upper bound for the spatial \(L_2\)-Sobolev regularity of the solution which restricts the order of convergence of uniform approximation methods.

MSC:

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60G51 Processes with independent increments; Lévy processes
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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