标题： 一个Brenier定理在$(\mathcal{P}_2 (\mathcal{P}_2(\mathbb{R}^d )), W_2 )$上的应用及适应运输的应用 显示英文标题

标题： A Brenier Theorem on $(\mathcal{P}_2 (\mathcal{P}_2(\mathbb{R}^d )), W_2 )$ and Applications to Adapted Transport

摘要： Brenier的基本定理通过凸函数的梯度，对度量$\mu, \nu$在$\mathbb{R}^d$上和二次距离成本的最优传输计划进行了表征。 特别是它保证了对于相对于Lebesgue测度绝对连续的度量存在最优传输映射。 我们的目标是为度量$P,Q$在$\mathcal{P}_2(\mathbb{R}^d)$上以及由平方Wasserstein距离$W_2^2(\mu, \nu)$给出的成本提供这个结果的一个版本。 我们根据Lions提升的凸性来表征优化器。 这是基于一个似乎具有独立兴趣的观察：泛函$\phi$的$c$变换，其中$c(\mu, \nu)$表示$\mu, \nu$的最大协方差，恰好对应于$\phi$的 Lions 提升的 Legendre 变换。 此外我们证明，对于典型的$P \in\mathcal{P}_2(\mathbb{R}^d)$，优化器是唯一的，并由一个传输映射给出。 在没有$\mathcal{P}_2(\mathbb{R}^d)$上规范参考测度的情况下，我们使用拓扑概念来精确描述“典型”。 具体来说，我们证明传输正则测度属于第二 Baire 类。 我们文章的一个特殊动机源于适应传输理论，其中适应 Wasserstein 距离为随机过程之间提供了适当的距离。 与其他度量相比，适应 Wasserstein 距离能够保持 Doob 分解、最优停止和随机控制问题的连续性。 基于我们对$\mathcal{P}_2(\mathbb{R}^d)$的结果，我们得到了适应性 Wasserstein 距离的第一个 Brenier 型定理。 显示英文摘要

摘要： Brenier's fundamental theorem characterizes optimal transport plans for measures $\mu, \nu$ on $\mathbb{R}^d$ and quadratic distance costs in terms of gradients of convex functions. In particular it guarantees the existence of optimal transport maps for measures which are absolutely continuous wrt Lebesgue measure. Our goal is to provide a version of this result for measures $P,Q$ on $\mathcal{P}_2(\mathbb{R}^d)$ and costs given by the squared Wasserstein distance $W_2^2(\mu, \nu)$. We characterize optimizers in terms of convexity of the Lions lift. This is based on an observation which seems to be of independent interest: the $c$-transform of a functional $\phi$, where $c(\mu, \nu)$ denotes maximal covariance of $\mu, \nu$ corresponds precisely to the Legendre transform of the Lions lift of $\phi$. Moreover we show that for typical $P \in\mathcal{P}_2(\mathbb{R}^d)$ the optimizer is unique and given by a transport map. In the absence of a canonical reference measure on $\mathcal{P}_2(\mathbb{R}^d)$ we use a topological notion to make `typical' precise. Specifically we show that the transport regular measures are of second Baire category. A particular motivation for our article stems from the theory of adapted transport where the adapted Wasserstein distance provides an adequate distance between stochastic processes. In contrast to other metrics, the adapted Wasserstein distance yields continuity of Doob-decomposition, optimal stopping and stochastic control problems. Based on our results for measures on $\mathcal{P}_2(\mathbb{R}^d)$ we obtain a first Brenier-type theorem for the adapted Wasserstein distance.