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The numerical methods for Stochastic PDEs studied so far in the literature
are predominantly based on uniform space time refinements. However, in the
deterministic world, there has been already shown that non-uniform refinement
strategies like adaptive wavelet-based methods for elliptic PDEs on bounded
Lipschitz domains $\mathcal{O}\subset\mathbb{R}^d$ can yield better convergence rates than uniform
alternatives. Thus, it is an immediate question whether adaptivity pays also
when dealing with Stochastic PDEs. In order to clarify this question one first of
all needs to analyse the regularity of the solutions using the so-called adaptivity
scale
$$\label{Cioica:eq:line}
B^\alpha_{\tau,\tau}(\mathcal{O}),\quad \frac{1}{\tau}=\frac{\alpha}{d}+\frac{1}{p},
$$
of Besov spaces to measure the smoothness with respect to the space variable
($p \geq 2$ is fixed). The regularity of a function in these Besov spaces determines
the convergence rate of the best $N$-term wavelet approximation and therefore
yields a benchmark for the convergence rate of adaptive wavelet methods.
In this talk, we first present recent results regarding the spatial regularity
of the solution to stochastic parabolic second order PDEs on general bounded
Lipschitz domains in the adaptivity scale \eqref{Cioica:eq:line} of Besov spaces. In particular,
we will present a general embedding of weighted Sobolev spaces into Besov
spaces. This is a key ingredient in order to obtain the desired results within
the framework of the analytic approach initiated by N.V. Krylov.
we move on to the analysis of the joint time-space regularity of the solution.
By a combination of the analytic approach with the semigroup approach
of Da Prato/Zabczyk and its recent extension by van Neerven/Veraar/Weis we
can establish an $L_q(L_p)$-regularity theory for SPDEs, where the integrability
parameters $q$ and $p$ in time and space, respectively, might differ. As a consequence
we can prove a time-space Hölder-Besov regularity result for the inhomogeneous
stochastic heat equation with additive noise on general bounded
Lipschitz domains.
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