A synthetic approach to comparison principles for variational problems, with applications to optimal transport

TL;DR

This paper introduces a synthetic variational framework for comparison principles in infinite-dimensional spaces, leveraging order-theoretic structures like submodularity and substitutability, with applications to optimal transport problems.

Contribution

It extends the concepts of submodularity and substitutability to infinite-dimensional spaces and applies them to derive comparison principles in optimal transport without regularity assumptions.

Findings

Duality between submodular functionals and substitutable conjugates

Comparison principles for Kantorovich potentials in various settings

Transport costs are shown to be substitutable, enabling analysis of JKO schemes

Abstract

We develop a synthetic, variational framework for deriving comparison principles in infinite-dimensional Banach spaces. Unlike traditional approaches that rely on the regularity of minimizers and Euler--Lagrange equations, our method exploits the order-theoretic structure of the energy. Central to our analysis is the notion of submodularity and its convex dual, substitutability, which we extend here to the infinite-dimensional setting. We prove a duality theorem establishing that a convex functional is submodular if and only if its conjugate is substitutable. We apply these results to problems in optimal transport, and derive comparison principles for Kantorovich potentials in standard, entropic, and unbalanced settings without requiring regularity assumptions on the cost or domain. Finally, we prove that general transport costs are substitutable, yielding comparison principles for JKO schemes driven by internal energies.