$L^1$-Uniqueness of the Fokker-Planck equation on a Riemannian manifold

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