Petru A. Cioica

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.

Petru A. Cioica, Kyeong-Hun Kim, Kijung Lee et al.

We investigate the regularity of linear stochastic parabolic equations with zero Dirichlet boundary condition on bounded Lipschitz domains $O \subset R^d$ with both theoretical and numerical purpose. We use N.V. Krylov's framework of stochastic parabolic weighted Sobolev spaces $\mathfrak{H}^{γ,q}_{p,θ}(O;T)$. The summability parameters p and q in space and time may differ. Existence and uniqueness of solutions in these spaces is established and the Hölder regularity in time is analysed. Moreover, we prove a general embedding of weighted Lp(O)-Sobolev spaces into the scale of Besov spaces $B^α_{τ,τ}(O), 1/τ=α/d+1/p, α> 0$. This leads to a Hölder-Besov regularity result for the solution process. The regularity in this Besov scale determines the order of convergence that can be achieved by certain nonlinear approximation schemes.