Brenier and Benamou-Brenier in Stochastic Mass Transport

Over the past decades, optimal transport has become a remarkably versatile mathematical framework used in fields ranging from physics and economics to finance and modern machine learning. At a high level, it offers a systematic way to compare different states or datasets and to determine how one can be transformed into another in an efficient manner. In mathematical terms, this means comparing probability distributions, and this perspective has revealed deep geometric structure and enabled new approaches to understanding complex systems. Causal optimal transport extends this perspective to stochastic processessystems that unfold over time and react to incoming information. In many real-world settings, one wants to compare different such processes while respecting the fact that decisions at any moment can only depend on what is known up to that point. This is essential in areas such as optimization and financial mathematics, where uncertainty evolves step by step and information arrives gradually. While classical optimal transport is now well understood and supported by a rich geometric theory, its causal counterpart is still far less developed. This project aims to close this gap. A central objective is to identify a new analogue of convexityone of the core structural ideas that underpins much of the classical theory and enables the characterization of optimal transport solutions. Establishing an appropriate counterpart in the causal setting opens the door to a deeper geometric understanding of adapted transport problems. By advancing these goals, the project will clarify how information constraints influence optimal decision-making in dynamic systems. It aims to provide mathematical foundations that parallel the classical theory while capturing the complexities of time-dependent uncertainty. In doing so, it will offer new tools for future research and applications across several fields where stochastic processes and information play a central role.