最优传输计划的定量稳定性、强制性（Coercivity）与唯一性
Quantitative Stability, Coercivity and Uniqueness of Optimal Transport Plans
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中文总结 AI 辅助
本文针对紧支撑概率测度，在Ahlfors正则条件下，定量证明了二次最优传输计划的稳定性、强制性和唯一性，并给出指数最优的反例。
中文摘要 AI 辅助
对于在 $\mathbb{R}^d$ 上支撑于固定紧集中的概率测度，我们证明了：在假设其中一个初始测度满足指数严格大于 $d-1$ 的上 Ahlfors 正则性条件时，二次最优传输计划在 Wasserstein 距离下对两个边际测度的扰动是定量稳定的。在相同假设下，我们还证明了关于最优传输映射的新颖定量稳定性结果。此外，我们证明了一个强制性定理，该定理指出：如果乘积空间上的任何概率测度具有相似的边际和与最优值相似的传输代价，那么它必须在定量意义上接近最优计划集合。最后，我们证明了一个定量唯一性定理，该定理作为 Brenier 定理的定量对应物。最优计划集合的 Wasserstein 直径由某个边际测度到满足唯一性的正则测度的距离所控制。这样，最优计划的“几乎唯一性”通过源测度的“几乎正则性”得以量化。我们提供了例子，证明所有估计的指数都是精确的（sharp）。
英文摘要
For probability measures on $\mathbb{R}^d$ supported in fixed compact sets, we prove that quadratic optimal transport plans are quantitatively stable in Wasserstein distance under perturbation of both marginal measures, assuming that one of the initial measures satisfies an upper Ahlfors regularity condition with exponent strictly greater than $d-1$. Under the same assumption, we also prove novel quantitative stability results for optimal transport maps. Furthermore, we prove a coercivity theorem which states that any probability measure on the product space must be quantitatively close to the set of optimal plans, if it has similar marginals and a similar transport cost to the optimal value. Finally, we prove a quantitative uniqueness theorem which acts as a quantitative counterpart to Brenier's theorem. The Wasserstein diameter of the set of optimal plans is controlled by the distance of one marginal measure to a regular measure for which uniqueness holds. In this way, "almost uniqueness" of optimal plans is quantified by the source measure being "almost regular". Examples are provided which prove that the exponents of all estimates are sharp.
发表机构
- CMAP École Polytechnique（巴黎综合理工学院 CMAP）
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