Some SDEs with distributional drift. Part II: Lyons-Zheng structure, Itô's formula and semimartingale characterization

In dimension 1 we study a martingale problem related to a parabolic PDE operator L with continuous (non-degenerate) diffusion term and with drift being the derivative of a continuous function. We state a necessary and sufficient condition on f and the solution X such that f(X) is a semimartingale. When X is a semimartingale, we also establish an Itô formula for f(X) under minimal assumptions. Particular attention is devoted to the case when L is close to divergence form.