We investigate contraction of the Wasserstein distances on \(\mathbb {R}^{d}\) under Gaussian smoothing. It is well known that the heat semigroup is exponentially contractive with respect to the Wasserstein distances on manifolds of positive curvature; however, on flat Euclidean space—where the heat semigroup corresponds to smoothing the measures by Gaussian convolution—the situation is more subtle. We prove precise asymptotics for the 2-Wasserstein distance under the action of the Euclidean heat semigroup, and show that, in contrast to the positively curved case, the contraction rate is always polynomial, with exponent depending on the moment sequences of the measures. We establish similar results for the p-Wasserstein distances for p≠ 2 as well as the χ² divergence, relative entropy, and total variation distance. Together, these results establish the central role of moment matching arguments in the analysis of measures smoothed by Gaussian convolution.

我们研究在高斯平滑下\（\ mathbb {R} ^ {d} \）上Wasserstein距离的收缩。众所周知，热半群相对于正曲率流形上的Wasserstein距离是指数收缩的。但是，在平坦的欧几里德空间上（热半群对应于通过高斯卷积平滑测度），情况就更加微妙了。我们证明了在欧几里得热半群的作用下2-Wasserstein距离的精确渐近性，并表明，与正曲率相比，收缩率始终是多项式的，其指数取决于测度的矩序列。对于p ≠2的p -Wasserstein距离以及χ ²发散，相对熵，和总偏差距离。总之，这些结果确立了矩匹配自变量在高斯卷积平滑测度分析中的核心作用。