合成曲率维数条件对传输距离指数的独立性
Independence of synthetic Curvature Dimension conditions on transport distance exponent
AI总结：
本文证明度量测度空间中基于不同距离幂次传输代价的曲率维数条件 $\mathrm{CD}_{p}(K,N)$ 在 $p>1$ 时相互等价，并揭示其统一纽带为 $L^{1}$-最优传输的针分解技术，同时建立了局部到整体性质。
摘要：
著名的 Lott-Sturm-Villani 度量测度空间理论提供了 Ricci 曲率下界 $K$ 与维数上界 $N$ 相结合的合成概念。他们的条件被称为曲率维数条件，记作 $\mathrm{CD}(K,N)$，其表述基于熵泛函沿 $W_{2}$-Wasserstein 测地线的修正位移凸性。我们证明选择平方距离函数作为传输代价不会影响该理论。将代价取为距离的 $p$ 次幂时的类似条件记作 $\mathrm{CD}_{p}(K,N)$，我们证明对于任意 $p>1$，$\mathrm{CD}_{p}(K,N)$ 都是等价条件——至少在测地线不分岔的空间中如此。我们证明所有这些表面上互不相关的 $\mathrm{CD}_{p}(K,N)$ 条件之间的纽带是与 $L^{1}$-最优传输问题相关联的针分解或局部化技术。我们还建立了 $\mathrm{CD}_{p}(K,N)$ 空间的局部到整体性质。
英文摘要：
The celebrated Lott-Sturm-Villani theory of metric measure spaces furnishes synthetic notions of a Ricci curvature lower bound $K$ joint with an upper bound $N$ on the dimension. Their condition, called the Curvature-Dimension condition and denoted by $\mathrm{CD}(K,N)$, is formulated in terms of a modified displacement convexity of an entropy functional along $W_{2}$-Wasserstein geodesics. We show that the choice of the squared-distance function as transport cost does not influence the theory. By denoting with $\mathrm{CD}_{p}(K,N)$ the analogous condition but with the cost as the $p^{th}$ power of the distance, we show that $\mathrm{CD}_{p}(K,N)$ are all equivalent conditions for any $p>1$ -- at least in spaces whose geodesics do not branch. We show that the trait d'union between all the seemingly unrelated $\mathrm{CD}_{p}(K,N)$ conditions is the needle decomposition or localization technique associated to the $L^{1}$-optimal transport problem. We also establish the local-to-global property of $\mathrm{CD}_{p}(K,N)$ spaces.
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