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 $\cO \subset \bR^d$ allowing Hardy inequality:   $$ \int_{\cO}|\rho^{-1}g|^2\,dx\leq C\int_{\cO}|g_x|^2 dx, \quad \forall g\in C^{\infty}_0(\cO), $$ where $\rho(x)=\text{dist}(x,\partial \cO)$.   Existence and uniqueness results are given in weighted Sobolev spaces $\frH^{\gamma}_{p,\theta}(\cO,T)$, where $p\in [2,\infty)$, $\gamma\in \bR$ 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. 
Kyeong-Hun Kim
math.PR