On a class of optimal transportation problems with infinitely many marginals

TL;DR

This paper establishes existence, uniqueness, and explicit characterization of solutions for a class of optimal transportation problems with infinitely many marginals, with applications in finance, physics, and PDE analysis.

Contribution

It introduces a novel framework for infinite marginal optimal transport, providing explicit solutions and broad applications in various scientific fields.

Findings

Proved existence and uniqueness of solutions

Derived an explicit formula for solutions

Applied results to finance and quantum physics

Abstract

We prove existence and uniqueness results for solutions to a class of optimal transportation problems with infinitely many marginals, supported on the real line. We also provide a characterization of the solution with an explicit formula. We then show that this result implies an infinite dimensional rearrangement inequality, and use it to derive upper bounds on solutions to parabolic PDE via the Feynman-Kac formula. This result can be used to provide model independent bounds on the prices of certain derivatives in mathematical finance. As an another application, we obtain refined phase-space bounds in quantum physics.