A weighted Sobolev space theory of parabolic stochastic PDEs on non-smooth domains

In this paper we study parabolic stochastic partial differential equations defined on arbitrary bounded domain $\text{cO}\subset {\text{bR}}^d$ allowing Hardy inequality: $${\int }_{\text{cO}}\mathrm{\mid }{\rho }^{-1}g{\mathrm{\mid }}^{2}dx\le C{\int }_{\text{cO}}\mathrm{\mid }{g}_x{\mathrm{\mid }}^{2}dx,\mathrm{\forall }g\in C_0^{\mathrm{\infty }}(\text{cO}),$$ where $\rho (x)=\text{dist}(x,\mathrm{\partial }\text{cO})$. Existence and uniqueness results are given in weighted Sobolev spaces ${\text{frH}}_{p,\theta }^{\gamma }(\text{cO},T)$, where $p\in [2,\mathrm{\infty })$, $\gamma \in \text{bR}$ is the number of derivatives of solutions and $\theta$ controls the boundary behavior of solutions. Furthermore several H"older estimates of the solutions are also obtained. It is allowed that the coefficients of the equations blow up near the boundary.