Remarks on curvature in the transportation metric
Remarks on curvature in the transportation metric
复制标题
关于运输度量中曲率的备注
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
A. Kolesnikov
中科院分区：
文献类型：
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作者：
B. Klartag;A. Kolesnikov
根据E. Calabi证明了任何具有自然Hessian度量的双曲仿射超球面都有一个非正的Ricci张量。仿射超球面可以描述为“双曲”复曲面Kähler-Einstein方程eΦ = detD 2 Φ在适当凸锥上的解的水平集。我们证明了这个定理的一个推广，证明了对于在适当凸域Ω上求解这个方程的每一个Φ，相应的度量测度空间（D2Φ，eΦdx）都有一个非正的Bakry-Émery张量。修改卡拉比计算，我们通过将张量极大值原理应用于巴克里-埃默里张量的加权拉普拉斯算子来获得这个结果。我们的计算进行了适应于任意目标和源措施的最优运输问题的广义框架。对于对数凹概率测度的最优输运问题，我们证明了一个三阶一致无量纲先验估计，其实质是二阶Caffarelli压缩定理，该定理在概率论中有着广泛的应用.
According to a classical result of E. Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a non-positive Ricci tensor. The affine hyperspheres can be described as the level sets of solutions to the “hyperbolic” toric Kähler–Einstein equation eΦ = detD2Φ on proper convex cones. We prove a generalization of this theorem by showing that for every Φ solving this equation on a proper convex domain Ω the corresponding metric measure space (D2Φ, eΦdx) has a non-positive Bakry–Émery tensor. Modifying the Calabi computations we obtain this result by applying the tensorial maximum principle to the weighted Laplacian of the Bakry–Émery tensor. Our computations are carried out in a generalized framework adapted to the optimal transportation problem for arbitrary target and source measures. For the optimal transportation of the log-concave probability measures we prove a third-order uniform dimension-free apriori estimate in the spirit of the second-order Caffarelli contraction theorem, which has numerous applications in probability theory.