The Wasserstein distance $\mathcal{W}_p$ is an important instance of an optimal transport cost. Its numerous mathematical properties as well as applications to various fields such as mathematical finance and statistics have been well studied in recent years. The adapted Wasserstein distance $\mathcal{A}\mathcal{W}_p$ extends this theory to laws of discrete time stochastic processes in their natural filtrations, making it particularly well suited for analyzing time-dependent stochastic optimization problems. While the topological differences between $\mathcal{A}\mathcal{W}_p$ and $\mathcal{W}_p$ are well understood, their differences as metrics remain largely unexplored beyond the trivial bound $\mathcal{W}_p\lesssim \mathcal{A}\mathcal{W}_p$. This paper closes this gap by providing upper bounds of $\mathcal{A}\mathcal{W}_p$ in terms of $\mathcal{W}_p$ through investigation of the smooth adapted Wasserstein distance. Our upper bounds are explicit and are given by a sum of $\mathcal{W}_p$, Eder's modulus of continuity and a term characterizing the tail behavior of measures. As a consequence, upper bounds on $\mathcal{W}_p$ automatically hold for $\mathcal{AW}_p$ under mild regularity assumptions on the measures considered. A particular instance of our findings is the inequality $\mathcal{A}\mathcal{W}_1\le C\sqrt{\mathcal{W}_1}$ on the set of measures that have Lipschitz kernels. Our work also reveals how smoothing of measures affects the adapted weak topology. In fact, we find that the topology induced by the smooth adapted Wasserstein distance exhibits a non-trivial interpolation property, which we characterize explicitly: it lies in between the adapted weak topology and the weak topology, and the inclusion is governed by the decay of the smoothing parameter.

翻译：Wasserstein距离$\mathcal{W}_p$是最优传输成本的一个重要实例。近年来，其众多数学性质以及在数理金融和统计学等领域的应用已得到深入研究。适应Wasserstein距离$\mathcal{A}\mathcal{W}_p$将该理论扩展到离散时间随机过程律在其自然滤波下的情形，使其特别适用于分析时间依赖的随机优化问题。尽管$\mathcal{A}\mathcal{W}_p$与$\mathcal{W}_p$之间的拓扑差异已得到充分理解，但作为度量而言，除了平凡上界$\mathcal{W}_p\lesssim \mathcal{A}\mathcal{W}_p$之外，二者的差异在很大程度上仍未得到探索。本文通过研究光滑适应Wasserstein距离，建立了用$\mathcal{W}_p$表示$\mathcal{A}\mathcal{W}_p$的上界，从而填补了这一空白。我们给出的上界是显式的，由$\mathcal{W}_p$、Eder连续模和刻画测度尾部行为的项之和构成。因此，在考虑测度满足温和正则性假设的条件下，$\mathcal{W}_p$的上界可自动推广至$\mathcal{AW}_p$。我们研究结果的一个特例是在具有Lipschitz核的测度集上成立的不等式$\mathcal{A}\mathcal{W}_1\le C\sqrt{\mathcal{W}_1}$。我们的工作还揭示了测度光滑化如何影响适应弱拓扑。事实上，我们发现由光滑适应Wasserstein距离诱导的拓扑展现出非平凡的插值性质，并对此进行了显式刻画：该拓扑介于适应弱拓扑与弱拓扑之间，其包含关系由光滑化参数的衰减速率所决定。