Hörmander Type Theorem and Maximum Principle for Stochastic PDEs

We shall first discuss the Hörmander theorem for general $I{t}^o$ processes and related (Kolmogrov) forward/backward stochastic PDEs, which may be beyond the scope of Markovian framework. Furthermore, we would also present the maximum principle for forward stochastic PDEs under a Hörmander type condition, which states the $L^p$ ($p\ge 2$) estimates for the time-space uniform norm of weak solutions.