  In this paper we derive quantitative uniqueness estimates at infinity for
solutions to an elliptic equation with unbounded drift in the plane. More
precisely, let $u$ be a real solution to $\Delta u+W\cdot\nabla u=0$ in
${\mathbf R}^2$, where $W$ is real vector and $\|W\|_{L^p({\mathbf R}^2)}\le K$
for $2\le p<\infty$. Assume that $\|u\|_{L^{\infty}({\mathbf R}^2)}\le C_0$ and
satisfies certain a priori assumption at $0$. Then $u$ satisfies the following
asymptotic estimates at $R\gg 1$ \[ \inf_{|z_0|=R}\sup_{|z-z_0|<1}|u(z)|\ge
\exp(-C_1R^{1-2/p}\log R)\quad\text{if}\quad 2<p<\infty \] and \[
\inf_{|z_0|=R}\sup_{|z-z_0|<1}|u(z)|\ge R^{-C_2}\quad\text{if}\quad p=2, \]
where $C_1>0$ depends on $p, K, C_0$, while $C_2>0$ depends on $K, C_0$ . Using
the scaling argument in [BK05], these quantitative estimates are easy
consequences of estimates of the maximal vanishing order for solutions of the
local problem. The estimate of the maximal vanishing order is a quantitative
form of the strong unique continuation property.
