Greens function for second order parabolic equations with singular lower order coefficients

We construct Greens functions for second order parabolic operators of the form $Pu=partial_t u-{rm div}({bf A} abla u+ boldsymbol{b}u)+ boldsymbol{c} cdot abla u+du$ in $(-infty, infty) times Omega$, where $Omega$ is an open connected set in $mathbb{R}^n$. It is not necessary that $Omega$ to be bounded and $Omega = mathbb{R}^n$ is not excluded. We assume that the leading coefficients $bf A$ are bounded and measurable and the lower order coefficients $boldsymbol{b}$, $boldsymbol{c}$, and $d$ belong to critical mixed norm Lebesgue spaces and satisfy the conditions $d-{rm div} boldsymbol{b} ge 0$ and ${rm div}(boldsymbol{b}-boldsymbol{c}) ge 0$. We show that the Greens function has the Gaussian bound in the entire $(-infty, infty) times Omega$.