AI 中文总结
该研究将Brenier-Strassen定理推广到非平坦CAT(κ)空间，证明了CAT(0)空间上有限二阶矩概率测度的凸序W₂投影存在唯一且由1-Lipschitz映射诱导，还建立了κ≥0时的局部化版本及一步重心鞅的Strassen型刻画。
AI 中文摘要
我们将关于凸序投影的Brenier-Strassen定理推广到具有上曲率界的非平坦空间。具体而言，对于完备可分CAT(0)空间上具有有限二阶矩的概率测度$μ$、$ν$，我们证明$μ$在凸序下被$ν$支配的概率测度集合上存在唯一的$W_2$投影$\bar{μ}$。此外，从$μ$到$\bar{μ}$的唯一最优耦合由一个1-Lipschitz映射诱导，且无需对$μ$施加任何绝对连续性假设。我们的证明将该投影问题等价于一个弱最优传输问题，其代价为到凸均值集合的距离平方。我们还在$κ\geq0$的CAT(κ)空间上建立了局部化版本，其中最优映射是1/2-Hölder连续的。最后，我们给出了紧CAT(0)空间上一步重心鞅的Strassen型刻画。
英文摘要
We extend the Brenier-Strassen theorem about projections in convex order to non-flat spaces with curvature bounded from above. Precisely, for probability measures $μ$, $ν$ of finite second moment on a complete separable CAT(0) space, we prove that $μ$ admits a unique W 2 -projection \barμ to the set of probability measures dominated by $ν$ in convex order. Moreover, the unique optimal coupling from $μ$ to \barμ is induced by a 1-Lipschitz map, without any absolute-continuity assumption on $μ$. Our proof identifies the projection problem with a weak optimal transport problem whose cost is the squared distance to the set of convex means. We also establish a localized version on CAT(kappa) spaces with kappa \geq 0, where the optimal map is 1/2-H{ö}lder continuous. Finally, we give a Strassen-type characterization of one-step barycentric martingales on proper CAT(0) spaces.
