标题： 完全凸函数，$L^2$-最优传输对于随机测度的分布，以及蒙日问题的解 显示英文标题

标题： Totally convex functions, $L^2$-Optimal transport for laws of random measures, and solution to the Monge problem

摘要： 我们研究在Kantorovich-Wasserstein空间$\mathcal{P}_2(\mathcal{P}_2(\mathrm{H}))$中的随机测度的定律的最优传输问题，该空间与一个希尔伯特空间$\mathrm{H}$（有限或无限维）相关，并且对应于由$\\mathcal{P}_2(\mathrm{H}).$中的平方Wasserstein度量引起的二次成本。尽管成本缺乏光滑性，空间$\mathcal{P}_2(\mathrm{H})$不是希尔伯特型的，并且底层Wasserstein度量引起的曲率失真，我们将展示如何在$\mathcal{P}_2(\mathcal{P}_2(\mathrm{H}))$中的随机测度层面上恢复将欧几里得最优传输问题与凸分析相联系的同样深刻而强大的结果，这些结果适用于$\mathcal{P}_2(\mathrm{H})$。 我们的方法依赖于完全凸泛函、它们的总次微分以及空间平方可积$\mathrm{H}$-值映射$L^2(\mathrm{Q},\mathbb{M};\mathrm{H}).$中的拉格朗日提升。 利用这些工具，我们确定了$\mathcal{P}_2(\mathcal{P}_2(\mathrm{H}))$中的一类自然正则测度，对于这些测度，OT 问题的蒙日表述具有唯一解，并且我们将证明这个类包括$\mathcal{P}_2(\mathrm{H})$中具有完整支撑的相关测度示例，这些测度来源于$L^2(\mathrm{Q},\mathbb{M};\mathrm{H}).$中非退化高斯测度的前向推导变换。 显示英文摘要

摘要： We study the Optimal Transport problem for laws of random measures in the Kantorovich-Wasserstein space $\mathcal{P}_2(\mathcal{P}_2(\mathrm{H}))$, associated with a Hilbert space $\mathrm{H}$ (with finite or infinite dimension) and for the corresponding quadratic cost induced by the squared Wasserstein metric in $\\mathcal{P}_2(\mathrm{H}).$ Despite the lack of smoothness of the cost, the fact that the space $\mathcal{P}_2(\mathrm{H})$ is not Hilbertian, and the curvature distortion induced by the underlying Wasserstein metric, we will show how to recover at the level of random measures in $\mathcal{P}_2(\mathcal{P}_2(\mathrm{H}))$ the same deep and powerful results linking Euclidean Optimal Transport problems in $\mathcal{P}_2(\mathrm{H})$ and convex analysis. Our approach relies on the notion of totally convex functionals, on their total subdifferentials, and their Lagrangian liftings in the space square integrable $\mathrm{H}$-valued maps $L^2(\mathrm{Q},\mathbb{M};\mathrm{H}).$ With these tools, we identify a natural class of regular measures in $\mathcal{P}_2(\mathcal{P}_2(\mathrm{H}))$ for which the Monge formulation of the OT problem has a unique solution and we will show that this class includes relevant examples of measures with full support in $\mathcal{P}_2(\mathrm{H})$ arising from the push-forward transformation of nondegenerate Gaussian measures in $L^2(\mathrm{Q},\mathbb{M};\mathrm{H}).$