  We consider uniformly parabolic equations and inequalities of second order in
the non-divergence form with drift
\[-u_{t}+Lu=-u_{t}+\sum_{ij}a_{ij}D_{ij}u+\sum b_{i}D_{i}u=0\,(\geq0,\,\leq0)\]
in some domain $Q\subset \mathbb{R}^{n+1}$. We prove growth theorems and the
interior Harnack inequality as the main results. In this paper, we will only
assume the drift $b$ is in certain Lebesgue spaces which are critical under the
parabolic scaling but not necessarily to be bounded. In the last section, some
applications of the interior Harnack inequality are presented. In particular,
we show there is a "universal" spectral gap for the associated elliptic
operator. The counterpart for uniformly elliptic equations of second order in
non-divergence form is shown in \cite{S10}.
