On time-dependent functionals of diffusions corresponding to divergence form operators

We consider additive functionals as a time and space-dependent function of a diffusion corresponding to nonhomogeneous uniformly elliptic divergence form operator. We show that if the function belongs to natural domain of strong solutions of PDEs then there is a version of this function such that additive functional is a continuous Dirichlet process for almost every starting points of diffusion and we describe the martingale and the zero-quadratic variation parts of its decomposition. We show also that under slightly stronger condition on the function this property holds for every starting point. Finally, we prove an extension of the It\^o formula.