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

Abstract

In this paper, we obtain a necessary and sufficient condition for $L^{\infty }$-uniqueness of Sturm-Liouville operator $a(x)\frac{d^2}{d{x}^2}+b(x)\frac{d}{dx}-V$ 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 $\LL^V:=Δ+b \cdot\nabla -V$ 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 $\LL^V$.

arXiv comment27 pages