Heat kernel bounds for parabolic equations with singular (form-bounded) vector fields

We consider Kolmogorov operator $$-\nabla \cdot a \nabla + b $$ with measurable uniformly elliptic matrix and prove Gaussian lower upper bounds on its heat kernel under minimal assumptions the vector field divergence $$\mathrm{div\,}b$$ . More precisely, we prove: (1) bound, provided that $$\mathrm{div\,}b \ge 0$$ , is in class of form-bounded fields (containing e.g. $$L^d$$ weak class, as well some are not even $$L_{\mathrm{loc}}^{2+\varepsilon }$$ $$\varepsilon >0$$ ); these assumptions, bound general invalid; (2) form-bounded, positive part Kato class; (3) bounds, (4) A priori large containing fields, class.