Dynamic characterization of barycentric optimal transport problems and their martingale relaxation

Paper: arXiv 2511.21287

Authors: Ivan Guo, Severin Nilsson, Johannes Wiesel

Abstract

We extend the Benamou-Brenier formula from classical optimal transport to weak optimal transport and show that the barycentric optimal transport problem studied by Gozlan and Juillet has a dynamic analogue. We also investigate a martingale relaxation of this problem, and relate it to the martingale Benamou-Brenier formula of Backhoff-Veraguas, Beiglböck, Huesmann and Källblad.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical extension of the Benamou-Brenier dynamic optimal transport formula to weak optimal transport and martingale relaxations. It is a purely theoretical probability/math finance paper with rigorous proofs but lacks empirical backtesting, numerical implementations, or real-world financial data application.

Research Flowchart

  flowchart TD
    A["Research Goal: Dynamic characterization of barycentric & weak optimal transport"] --> B["Inputs: Initial & final probability distributions, Cost functions"]
    B --> C["Extend Classical Benamou-Brenier Formula to Weak OT"]
    C --> D["Formulate Dynamic Barycentric Optimal Transport"]
    D --> E["Apply Martingale Relaxation & Stochastic Control"]
    E --> F["Outcomes: Duality results & link to Martingale Benamou-Brenier formula"]