René L. Schilling

Petru A. Cioica, Stephan Dahlke, Stefan Kinzel et al.

We use the scale of Besov spaces B^α_{τ,τ}(O), α>0, 1/τ=α/d+1/p, p fixed, to study the spatial regularity of the solutions of linear parabolic stochastic partial differential equations on bounded Lipschitz domains O\subset R^d. The Besov smoothness determines the order of convergence that can be achieved by nonlinear approximation schemes. The proofs are based on a combination of weighted Sobolev estimates and characterizations of Besov spaces by wavelet expansions.

Mihály Kovács, Felix Lindner, René L. Schilling

We present an abstract framework to study weak convergence of numerical approximations of linear stochastic partial differential equations driven by additive Lévy noise. We first derive a representation formula for the error which we then apply to study space-time discretizations of the stochastic heat equation, a Volterra-type integro-differential equation, and the wave equation as examples. For twice continuously differentiable test functions with bounded second derivative (with an additional condition on the second derivative for the wave equation) the weak rate of convergence is found to be twice the strong rate. The results extend earlier work by two of the authors as we consider general square-integrable infinite-dimensional Lévy processes and do not require boundedness of the test functions and their first derivative. Furthermore, the present framework is applicable to both hyperbolic and parabolic equations, and even to stochastic Volterra integro-differential equations.

Felix Lindner, René L. Schilling

Considering a linear parabolic stochastic partial differential equation driven by impulsive space time noise, dX_t+AX_t dt= Q^{1/2}dZ_t, X_0=x_0\in H, t\in [0,T], we approximate the distribution of X_T. (Z_t)_{t\in[0,T]} is an impulsive cylindrical process and Q describes the spatial covariance structure of the noise; Tr(A^{-α})<\infty for some α>0 and A^βQ is bounded for some β\in(α-1,α]. A discretization (X_h^n)_{n\in\{0,1,...,N\}} is defined via the finite element method in space (parameter h>0) and a θ-method in time (parameter Δt=T/N). For ϕ\in C^2_b(H;R) we show an integral representation for the error |Eϕ(X^N_h)-Eϕ(X_T)| and prove that |Eϕ(X^N_h)-Eϕ(X_T)|=O(h^{2γ}+(Δt)^γ) where γ<1-α+β.