  In this paper, we obtain a necessary and sufficient condition for
$L^{\infty}$-uniqueness of Sturm-Liouville operator $a(x)\frac{d^2}{dx^2} +
b(x) \frac d{dx} -V$ on an open interval of $\rr$, which is equivalent to the
$L^1$-uniqueness of the associated Fokker-Planck equation. For a general
elliptic operator $\LL^V:=\Delta +b \cdot\nabla -V$ on a Riemannian manifold,
we obtain sharp sufficient conditions for the $L^1$-uniqueness of the
Fokker-Planck equation associated with $\LL^V$, via comparison with a
one-dimensional Sturm-Liouville operator. Furthermore the $L^1$-Liouville
property is derived as a direct consequence of the $L^\infty$-uniqueness of
$\LL^V$.
