AI 中文总结
本文证明Wasserstein空间在垂直与水平两种几何下均具有Radon-Nikodym性质，并由此推导变分原理、向量测度定理及鞅收敛结果，同时引入水平鞅并刻画其为L^p值鞅的分布。
AI 中文摘要
Wasserstein空间承载着两种自然几何：一种是由符号测度的线性结构继承而来的垂直几何，另一种是由耦合诱导的水平几何。我们证明Radon-Nikodym性质在这两种几何中均成立。在垂直几何中，$P_p(R^d)$的有界子集在一个本身不满足Radon-Nikodym性质的Banach空间中是可齿的；我们由此推导出变分原理、向量测度的Radon-Nikodym定理以及鞅收敛结果。在水平几何中，我们证明Radon-Nikodym性质是从随机变量空间的性质继承而来的。我们引入水平鞅，并证明它们恰好是$L^p$值鞅的分布。
英文摘要
Wasserstein spaces carry two natural geometries: a vertical one, inherited from the linear structure of signed measures, and a horizontal one, induced by couplings. We prove that a Radon-Nikodym property holds in both. In the vertical geometry, bounded subsets of $P_p(R^d)$ are dentable in an ambient Banach space which itself fails the Radon-Nikodym property; we derive variational principles, Radon-Nikodym theorems for vector measures and martingale convergence results. In the horizontal geometry, we show that a Radon-Nikodym property is inherited from the one of spaces of random variables. We introduce horizontal martingales and prove that they are exactly the laws of $L^p$ -valued martingales.
