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arXiv · 1307.7473

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

Bin Qian·Liming Wu
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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 $\LL^V:=Δ+b \cdot\nabla -V$ 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 $\LL^V$.

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Bin Qian, Liming Wu. 2013-07-29. $L^1$-Uniqueness of the Fokker-Planck equation on a Riemannian manifold. https://arxiv.org/abs/1307.7473

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