We provide a version for lower probabilities of Monge's and Kantorovich's optimal transport problems. We show that, when the lower probabilities are the lower envelopes of $\epsilon$-contaminated sets, then our version of Monge's, and a restricted version of our Kantorovich's problems, coincide with their respective classical versions. We also give sufficient conditions for the existence of our version of Kantorovich's optimal plan, and for the two problems to be equivalent. As a byproduct, we show that for $\epsilon$-contaminations the lower probability versions of Monge's and Kantorovich's optimal transport problems need not coincide. The applications of our results to Machine Learning and Artificial Intelligence are also discussed.

翻译：我们提出了针对下概率的Monge与Kantorovich最优传输问题的版本。研究表明，当下概率为$\epsilon$污染集的下包络时，我们所构建的Monge问题版本以及受限的Kantorovich问题版本将分别与其经典形式等价。我们同时给出了所构建的Kantorovich最优传输方案存在性的充分条件，以及两类问题等价的充分条件。作为推论，我们证明了对于$\epsilon$污染情形，下概率版本的Monge与Kantorovich最优传输问题未必等价。本文还探讨了相关结果在机器学习与人工智能领域的应用前景。