L-1-Uniqueness of the Fokker-Planck Equation on a Riemannian Manifold

摘要

In this paper, we obtain a necessary and sufficient condition for L-infinity-uniqueness of Sturm-Liouville operator a(x) d2/dx2 + b(x) d dx - V on an open interval of R, which is equivalent to the L1-uniqueness of the associated Fokker-Planck equation. For a general elliptic operator L-V := Delta + b center dot del - V on a Riemannian manifold, we obtain sharp sufficient conditions for the L1-uniqueness of the Fokker-Planck equation associated with L-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-infinity-uniqueness of L-V.