发表机构
École Normale Supérieure, Rennes（雷恩高等师范学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究 Lévy 律在弱扰动下的 Wasserstein 距离二阶展开，通过方差加权特征微分获得一致展开，并给出二次网格速率、度量速度及响应线性规划等结果。
AI 中文摘要
等大小的弱扰动可产生不同阶数的终端 Wasserstein 误差。对于高斯平滑的无穷可分律，我们相对于方差加权的 Lévy 特征对分布函数进行微分，包括由移动原子生成的分布方向。我们获得了对 Lipschitz 检验一致的一阶和二阶展开。当一阶响应非零时，高斯解析性确定了 Wasserstein-1 距离的带符号二阶系数。当一阶响应为零时，有限的二阶位移矩产生二次响应，且对于非零位移离散度严格为正。对于局部平衡重网格，误差与网格对齐方差相当，从而对紧支撑密度和非对齐原子序列给出尖锐的二次网格速率。我们推导了沿可容许非原子曲线的度量速度和长度，以及一个基于响应的线性规划，该规划具有经过认证的 oracle 间隙和一致的求积。我们还建立了尖锐的消失平滑过渡和多元二阶展开。补充材料处理状态依赖的响应和进一步的稳定性估计。
英文摘要
Weak perturbations of equal size can produce terminal Wasserstein errors of different orders. For Gaussian-smoothed infinitely divisible laws, we differentiate distribution functions with respect to variance-weighted Lévy characteristics, including distributional directions generated by moving atoms. We obtain first- and second-order expansions uniform over Lipschitz tests. Gaussian analyticity identifies the signed second coefficient of the Wasserstein-1 distance when the first response is nonzero. When it vanishes, a finite second displacement moment yields a quadratic response, strictly positive for nonzero displacement dispersion. For local balanced remeshing, the error is comparable to the grid-alignment variance, giving sharp quadratic grid rates for compactly supported densities and nonaligned atomic sequences. We derive metric speed and length along admissible non-atomic curves and a response-based linear program with a certified oracle gap and consistent quadrature. We also establish the sharp vanishing-smoothing transition and a multivariate second-order expansion. The Supplement treats state-dependent responses and further stability estimates.
Comments62 pages total (33-page main article and 29-page supplementary material)
