In this talk, we explore Stochastic Differential Equations (SDEs) and Partial Differential Equations (PDEs) with rough coefficients, focusing on the interplay between analysis and stochastic analysis. Starting from the Ornstein-Uhlenbeck process as a key example, we investigate the role of semigroups and invariant measures in general time-homogeneous Itô-SDEs. Furthermore, based on general second-order partial differential operators, we develop an analytic framework for constructing (infinitesimally) invariant measures and regularized semigroups. This enables us to construct solutions to the corresponding SDEs with a potential candidate for an invariant measure, or even with prescribed invariant measure. The presented approach naturally extends to more challenging situations. For instance, it applies to the study of degenerate SDEs, which illustrates the fundamental role of semigroups and invariant measures in both SDEs and PDEs.