Optimal transport provides a principled mathematical framework for quantifying dissimilarity between probability distributions. While its theory and computational methods are by now well established when the distributions are over the reals, their extensions to stochastic processes raise a number of largely open computational, statistical, and structural questions. Recent work on discrete time-space Markov processes now provides a framework for formulating and efficiently computing optimal transport problems between such processes, even without exact knowledge of the underlying dynamics. This project aims to refine and enrich this perspective by studying optimal transport for stochastic processes, from both theoretical and computational viewpoints. The central goals will be to develop efficient algorithms with theoretical guarantees and to investigate extensions to more general settings, including large or infinite state spaces. More broadly, the project seeks to understand how optimal transport can provide new tools for comparing, analyzing, and learning of stochastic processes. The project will also explore the potential application of these methods in different areas of Machine Learning, with particular interest in Reinforcement Learning.
