Kolmogorov operator with the vector field in Nash class

We consider divergence-form parabolic equation with measurable uniformly elliptic matrix and the vector field in a large class containing, particular, fields $L^p$, $p>d$, as well some that are not even $L_{\loc}^{2+\varepsilon}$, $\varepsilon>0$. establish Hölder continuity of bounded soutions, sharp two-sided Gaussian bound on heat kernel, Harnack inequality.