On \(L_p\)-theory of stochastic partial differential equations of divergence form in \(C^1\) domains
The author deals with an \(L^p\)-theory of stochastic partial differential equations of the form \[ du=(D_i(a^{ij}u_{x^j}+b^iu+f^i)+\bar{b}^i u_{x^i}+c u+ \bar{f})dt+(\sigma^{ik}u_{x^i} + \nu^k u+g^k)dw^k_t, \] where \(x\) belongs to some open set in \(\mathbb{R}^d\), \(t\geq 0\), and \((w^k_t)_{k\geq 1}\) are independent one-dimensional Wiener processes. The coefficients of the above equation are random functions depending both on the time and space variable. Existence and uniqueness of solution in a certain stochastic Banach space is studied. The techniques are based on Sobolev spaces with weights allowing the derivatives of the solutions to blow up near the boundary. Hölder continuity of the solution in \(t\) and \(x\) is also studied. The main motivation of this setting comes from \textit{N. V. Krylov} and \textit{S. V. Lototsky} [SIAM J. Math. Anal. 31, No. 1, 19--33 (1999; Zbl 0943.60047)].
scientific article; zbMATH DE number 2137176