含奇异基测度的概率测度空间上的输运型度量
Transport type metrics on the space of probability measures involving singular base measures
摘要：
本文在定义域$X \subseteq \mathbb{R}^m$上的概率测度$\mathcal P(X)$的集合上发展了一个度量理论，我们称之为基于$ν$的Wasserstein度量，记为$W_ν$.这个度量是基于广义测地线的概念相对于一个基本措施$ν$略有改进，特别是当$ν$是奇异的情况下，相对于$m$维勒贝格措施，它也是密切相关的线性化最佳运输的概念。$ν$为基础的Wasserstein度量定义的迭代变分问题，涉及最佳运输到$ν$，我们还描述了它在经典的Wasserstein距离之间的条件概率和通过限制某些多边际最佳运输问题的积分。当我们改变基测度$ν$时，基于$ν$的Wasserstein度量在通常的二次Wasserstein距离和与当$ν$充分正则时获得的唯一定义的广义测地线相关联的度量之间插值。当$ν$集中于$\mathbb{R}^m$的一个低维子流形时，我们证明了基于$ν$的Wasserstein距离定义中的变分问题有唯一解.我们建立测地凸的通常类的泛函和一组源措施$μ$，使$μ$和$ν$之间的最佳运输满足广义嵌套条件的加强介绍了\cite{McCannPass20}。最后，我们介绍了轻微的变化的对偶度量上述为了证明收敛的迭代方案，以解决变分问题中出现的博弈论。
英文摘要：
We develop the theory of a metric, which we call the $ν$-based Wasserstein metric and denote by $W_ν$, on the set of probability measures $\mathcal P(X)$ on a domain $X \subseteq \mathbb{R}^m$. This metric is based on a slight refinement of the notion of generalized geodesics with respect to a base measure $ν$ and is relevant in particular for the case when $ν$ is singular with respect to $m$-dimensional Lebesgue measure; it is also closely related to the concept of linearized optimal transport. The $ν$-based Wasserstein metric is defined in terms of an iterated variational problem involving optimal transport to $ν$; we also characterize it in terms of integrations of classical Wasserstein distance between the conditional probabilities and through limits of certain multi-marginal optimal transport problems. As we vary the base measure $ν$, the $ν$-based Wasserstein metric interpolates between the usual quadratic Wasserstein distance and a metric associated with the uniquely defined generalized geodesics obtained when $ν$ is sufficiently regular. When $ν$ concentrates on a lower dimensional submanifold of $\mathbb{R}^m$, we prove that the variational problem in the definition of the $ν$-based Wasserstein distance has a unique solution. We establish geodesic convexity of the usual class of functionals and of the set of source measures $μ$ such that optimal transport between $μ$ and $ν$ satisfies a strengthening of the generalized nestedness condition introduced in \cite{McCannPass20}.We finally introduce a slight variant of the dual metric mentioned above in order to prove convergence of an iterative scheme to solve a variational problem arising in game theory.
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