L¹-Uniqueness of the Fokker-Planck equation on a Riemannian manifold

2013/07/29 by Bin Qian, Liming Wu, Qian, Bin +1
Mathematics · #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.CA #math.DG #msc:34B24 #msc:53C21

paper · pdf · doi:10.48550/arxiv.1307.7473

27 pages

arxiv created 2013/07/29 · arxiv updated 2013/07/30

Abstract

In this paper, we obtain a necessary and sufficient condition for L^∞-uniqueness of Sturm-Liouville operator a(x)(d²)/(dx²) + b(x) \frac ddx -V on an open interval of \rr, which is equivalent to the L¹-uniqueness of the associated Fokker-Planck equation. For a general elliptic operator \LL^V:=Δ+b ⋅∇ -V on a Riemannian manifold, we obtain sharp sufficient conditions for the L¹-uniqueness of the Fokker-Planck equation associated with \LL^V, via comparison with a one-dimensional Sturm-Liouville operator. Furthermore the L¹-Liouville property is derived as a direct consequence of the L^∞-uniqueness of \LL^V.

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