On the $L_q(L_p)$-regularity and Besov smoothness of stochastic parabolic equations on bounded Lipschitz domains

Abstract

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.

Singular behavior of the solution to the stochastic heat equation on a polygonal domain

“…We are in particular interested in an explicit bound for the L 2 (O)-Sobolev regularity of u, which is closely connected to the order of convergence that can be achieved by uniform numerical approximation methods if the error is measured in L 2 (O). In this respect, our result complements the Besov regularity results in [2], [3], which are related to the order of convergence for non-uniform, adaptive approximation methods. We refer to [7] or [2, Section 1] for details on the connection between regularity and approximation.…”

Stochastic maximal regularity for rough time-dependent problems

“…[35,[55][56][57][58]60,61,63,74] where also the case of smooth domains has been considered, and later to e.g. [18][19][20]59,81] where the case of non-smooth domains is investigated. In the above mentioned results one uses L p -integrability in space, time and .…”

Stochastic Integration in Banach Spaces – a Survey

“…Applications of the theory of stochastic integration in UMD spaces have been worked out in a number of papers; see [12,13,18,21,22,24,30,57,56,58,78,80,81,93,96] and the references therein. Here we will limit ourselves to the maximal regularity theorem for stochastic convolutions from [81] which is obtained by combining Theorem 7.1 and 7.3 below, and which crucially depends on the sharp two-sided inequality of Theorem 5.5.…”