On SPDEs with variable coefficients in one space dimension (Q1876616)

This article deals with \(L^p\)-theory of Itô stochastic partial differential equations of the type \[ du=(au_{xx}+bu_x+cu+f)dt+(\sigma u_x+\nu u + g)\,dW \] on the bounded interval \([0,1]\). The aim of the paper is to prove existence and uniqueness in spaces of weighted Sobolev spaces, that allow for a blow up of the derivative near the boundary. This takes into account highly oscillatory \(a\), for instance \(a=2+\sin(| \ln(x)| ^\alpha)\) for \(\alpha\in(0,1)\). The authors show that in case of the interval the results of \textit{S. V. Lototsky} [Stochastics Stochastics Rep. 68, No. 1--2, 145--175 (1999; Zbl 0944.60076)] can be obtained in a different way. Furthermore, they relax continuity requirements.