Preprint des Projektes: SFB 701: Spektrale Strukturen und Topologische Methoden in der Mathematik - Projekt A9

Dynamics and asymptotic behaviour of stochastic evolution systems

13-078 Adam Andersson, Raphael Kruse, Stig Larsson.
Duality in refined Watanabe-Sobolev spaces and weak approximation of SPDE

In this paper we introduce a new family of refined Watanabe- Sobolev spaces that capture in a fine way integrability in time of the Malliavin derivative. We consider duality in these spaces and derive a Burkholder type inequality in a dual norm. The theory we develop allows us to prove weak convergence with essentially optimal rate for numerical approximations in space and time of semilinear parabolic stochastic evolution equations driven by Gaussian additive noise. In particular, we combine Galerkin finite element methods with a backward Euler scheme in time. The method of proof does not rely on the use of the Kolmogorov equation or the Ito formula and is therefore in nature non-Markovian. With this method polynomial growth test functions with mild smoothness assumptions are allowed, meaning in particular that we prove convergence of arbitrary moments with essentially optimal rate. Our Gronwall argument also yields weak error estimates which are uniform in time without any additional effort.