球面Hellinger-Kantorovich梯度流
Spherical Hellinger-Kantorovich gradient flows
摘要：
本文研究了一类退化的非线性Fokker-Planck型抛物方程，该方程可以看作是关于最近引入的球面Hellinger-Kantorovich距离的梯度流.驱动熵不被假定为测地线凸的。我们证明了问题的可解性和熵-熵产生不等式，这意味着指数收敛到平衡点。作为推论，我们得到了Wasserstein梯度流的一些相关结果。我们还推导出运输不等式的Talagrand，奥托和Villani的球形和圆锥Hellinger-Kantorovich距离的精神。
英文摘要：
We study nonlinear degenerate parabolic equations of Fokker-Planck type which can be viewed as gradient flows with respect to the recently introduced spherical Hellinger-Kantorovich distance. The driving entropy is not assumed to be geodesically convex. We prove solvability of the problem and the entropy-entropy production inequality, which implies exponential convergence to the equilibrium. As a corollary, we obtain some related results for the Wasserstein gradient flows. We also deduce transportation inequalities in the spirit of Talagrand, Otto and Villani for the spherical and conic Hellinger-Kantorovich distances.
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