The convergence rate of best N-term wavelet approximation is determined by the regularity of the target function in a special scale of Besov spaces---the so called non-linear approximation scale. We use this scale to measure the spatial smoothness of the solution to second order stochastic partial differential equations (SPDEs) on general bounded Lipschitz domains. Furthermore, we investigate the Hölder regularity of the paths of the solution to the stochastic heat equation, considered as a stochastic process taking values in the Besov spaces from the non-linear approximation scale.
First we prove a general embedding of the weighted Sobolev spaces used by N.V. Krylov and co-authors for the analysis of SPDEs into the Besov spaces from the non-linear approximation scale. This embedding shows that, up to a certain amount, the regularity analysis in the non-linear approximation scale can be traced back to the analysis of the weighted Sobolev regularity. As a consequence we can prove high spatial Besov regularity of the solution process in the non-linear approximation scale by using the weighted Sobolev space theory for SPDEs on bounded Lipschitz domains, recently developed by K.-H. Kim.
Our analysis of the time-space regularity of the stochastic heat equation is based on the following strategy. Firstly, we establish existence of a solution in the space of q-integrable stochastic processes taking values in the weighted Lp-Sobolev spaces mentioned above of sufficiently high smoothness. The integrability parameters "q" and "p" in time and space, respectively, may differ. Our proof relies on a combination of the semigroup approach and the analytic approach to SPDEs. From the semigroup approach we obtain existence of a solution with low weighted Sobolev regularity. Using techniques from the analytic approach we can lift this regularity, if we can increase the regularity of the free terms. Secondly, we present a result on the Hölder regularity of the paths of the solution, considered as a stochastic process taking values in weighted Sobolev spaces. From this we can extract a Hölder-Besov regularity result by using the embedding of weighted Sobolev spaces into Besov spaces from the non-linear approximation scale mentioned above.
This is joint work with Kyeong-Hun Kim (Korea University), Kijung Lee (Ajou University) and Felix Lindner (TU Kaiserslautern).