论文标题
通过熵公式在球体上运输
Transportation on spheres via an entropy formula
论文作者
论文摘要
该论文证明了运输不平等，以实现瓦斯恒星指标的概率措施，这些措施是关于地球距离的力量的成本函数的运输不平等。令$μ$为$dμ= e^e^{ - u（x）} dx $的球体$ {\ bf s}^n $的可能性度量然后，任何概率测量有限的相对熵相对于$μ$满足$ {\ hbox {ent}}}（ν\ mid -μ）\ geq（κ_U/2）w_2（ν，μ）^2 $。证明使用明确的公式对相对熵，该公式在连接和紧凑的$ c^\ indy $ spooth riemannian歧管没有边界的情况下也有效。此熵公式的变化给出了Lichnérowicz的积分。
The paper proves transportation inequalities for probability measures on spheres for the Wasserstein metrics with respect to cost functions that are powers of the geodesic distance. Let $μ$ be a probability measure on the sphere ${\bf S}^n$ of the form $dμ=e^{-U(x)}dx$ where $dx$ is the rotation invariant probability measure, and $(n-1)I+{\hbox{Hess}}\,U\geq {κ_U}I$, where $κ_U>0$. Then any probability measure $ν$ of finite relative entropy with respect to $μ$ satisfies ${\hbox{Ent}}(ν\midμ) \geq (κ_U/2)W_2(ν, μ)^2$. The proof uses an explicit formula for the relative entropy which is also valid on connected and compact $C^\infty$ smooth Riemannian manifolds without boundary. A variation of this entropy formula gives the Lichnérowicz integral.
