Abstract
In this paper, we obtain a necessary and sufficient condition for L∞-uniqueness of Sturm-Liouville operator \(a(x)\frac {d^{2}}{dx^{2}} + b(x) \frac d{dx} -V\) on an open interval of \(\mathbb {R}\), which is equivalent to the L1-uniqueness of the associated Fokker-Planck equation. For a general elliptic operator \(\mathcal {L}^{V}:={\Delta } +b \cdot \nabla -V\) on a Riemannian manifold, we obtain sharp sufficient conditions for the L1-uniqueness of the Fokker-Planck equation associated with \(\mathcal {L}^{V}\), 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 \(\mathcal {L}^{V}\).
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The author would like to express his sincere thanks to the anonymous referee for his/her valuable comment. He also acknowledges the financial support by Qing Lan Project and by National Science Funds of China No. 11671076.
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Qian, B., Wu, L. L1-Uniqueness of the Fokker-Planck Equation on a Riemannian Manifold. Potential Anal 51, 603–626 (2019). https://doi.org/10.1007/s11118-018-9727-1
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DOI: https://doi.org/10.1007/s11118-018-9727-1