We study optimal transport problems in which finite-valued quantities of interest evolve dynamically over time in a stationary fashion. Mathematically, this is a special case of the general optimal transport problem in which the distributions under study represent stationary processes and the cost depends on a finite number of time points. In this setting, we argue that one should restrict attention to stationary couplings, also known as joinings, which have close connections with long run average cost. We introduce estimators of both optimal joinings and the optimal joining cost, and we establish their consistency under mild conditions. Under stronger mixing assumptions we establish finite-sample error rates for the same estimators that extend the best known results in the iid case. Finally, we extend the consistency and rate analysis to an entropy-penalized version of the optimal joining problem.

翻译：我们研究的是最佳运输问题,在这些问题中,定值定值的利息数量会以固定的方式随时间动态地演变。从数学角度讲,这是一般最佳运输问题的一个特例,所研究的分布代表着固定过程,成本取决于一定的时间点。在这一背景下,我们主张,应当限制对固定的联结(又称连结)的关注,这些联结与长期平均成本有着密切的联系。我们引入了最佳联结和最佳联结成本的估测器,我们在温和条件下建立一致性。在较强的混合假设下,我们为扩大已知最佳联结问题最佳结果的同一估测器制定了限定的差错率。最后,我们将一致性和率分析扩大到最佳联结问题的最佳配方版本。