Besov regularity of stochastic partial differential equations on bounded Lipschitz domains
This thesis is concerned with the regularity of (semi-)linear second order parabolic stochastic partial differential equations (SPDEs, for short) of Itô type on bounded Lipschitz domains. The so-called adaptivity scale of Besov spaces is used to measure the regularity of the solution with respect to...
Reine und Angewandte Mathematik
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|Summary:||This thesis is concerned with the regularity of (semi-)linear second order parabolic stochastic partial differential equations (SPDEs, for short) of Itô type on bounded Lipschitz domains. The so-called adaptivity scale of Besov spaces is used to measure the regularity of the solution with respect to the space variable. It determines the convergence rate of the so-called best m-term wavelet approximation, which is the benchmark for modern adaptive numerical methods based on wavelet bases or frames. The regularity with respect to the time variable is measured in the classical Hölder-norm.
The analysis is put into the framework of the analytic approach for SPDEs initiated by Nicolai V. Krylov. Recent results by Kyeong-Hun Kim regarding the spatial weighted Sobolev regularity of the solutions to SPDEs on non-smooth domains (DOI:10.1007/s10959-012-0459-7) are the starting point of the investigations. General embeddings of weighted Sobolev spaces into the classical Sobolev spaces and into the Besov spaces from the adaptivity scale are proven. These embeddings together with a generalization of Kim's results to a class of semi-linear SPDEs yield the desired spatial Besov regularity results. In particular, it is shown that in specific situations the spatial Besov regularity of the solution in the adaptivity scale is generically higher than its classical Sobolev regularity. As it is well-known from approximation theory, this indicates that in many cases adaptive wavelet methods for solving SPDEs should be used instead of uniform alternatives. It is worth noting that the aforementioned embeddings are proven independently of the SPDE context and are relevant also in other mathematical fields.
In order to prove space time regularity of the solution, techniques from the analytic approach are combined with results obtained from the semigroup approach of Da Prato/Zabczyk (ISBN:9780521059800). This procedure yields an Lq(Lp)-theory for the heat equation with additive noise on general bounded Lipschitz domains. The integrability parameter q with respect to the time variable can be chosen to be strictly greater than the spatial integrability parameter p. As a consequence, Hölder-Besov regularity of the solution can be established.|