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

TL;DR

This paper establishes necessary and sufficient conditions for the $L^1$-uniqueness of the Fokker-Planck equation on Riemannian manifolds, linking it to Sturm-Liouville operators and deriving the $L^1$-Liouville property.

Contribution

It provides sharp criteria for $L^1$-uniqueness of the Fokker-Planck equation on manifolds, extending previous results through comparison with Sturm-Liouville operators.

Findings

Established necessary and sufficient conditions for $L^{ ext{infinity}}$-uniqueness.

Derived sharp sufficient conditions for $L^1$-uniqueness on Riemannian manifolds.

Connected $L^1$-uniqueness to the $L^1$-Liouville property.

Abstract

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 $\LL^V$.