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In this article we present a W n 2 -theory of stochastic parabolic partial differential systems. In particular, we focus on non-divergent type. The space domains we consider are R d , R d + and eventually general bounded C 1 -domains O. By the nature of stochastic parabolic equations we need weighted Sobolev spaces to prove the existence and the uniqueness. In our choice of spaces we allow the derivatives of the solution to blow up near the boundary and moreover the coefficients of the systems are allowed to oscillate to a great extent or blow up near the boundary.

We investigate the regularity of linear stochastic parabolic equations with zero Dirichlet boundary condition on bounded Lipschitz domains O ⊆ R d with both theoretical and numerical purpose. We use N.V. Krylov's framework of stochastic parabolic weighted Sobolev spaces 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 L p (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.

In this paper we present a weighted Lp-theory of second-order parabolic partial differential equations defined on C 1 domains. The leading coefficients are assumed to be measurable in time variable and have VMO (vanishing mean oscillation) or small BMO (bounded mean oscillation) with respect to space variables, and lower order coefficients are allowed to be unbounded and to blow up near the boundary. Our BMO condition is slightly relaxed than the others in the literature.

We establish existence, uniqueness, and arbitrary order Sobolev regularity results for the second order parabolic equations with measurable coefficients defined on the conic domains D of the typeWe obtain the regularity results by using a system of mixed weights consisting of appropriate powers of the distance to the vertex and of the distance to the boundary. We also provide the sharp ranges of admissible powers of the distance to the vertex and to the boundary.