A synthetic approach to comparison principles for variational problems, with applications to optimal transport
Flavien Léger, Maxime Sylvestre
TL;DR
The paper develops a synthetic, order-theoretic framework for deriving comparison principles in infinite-dimensional variational problems by leveraging submodularity in the primal space and substitutability in the convex dual. A duality theorem shows that a convex functional is submodular if and only if its conjugate is substitutable, enabling principled comparison principles that hold without regularity or uniqueness assumptions. The framework is then applied to optimal transport in standard, entropic, and unbalanced settings, yielding Kantorovich-potential comparisons and extending to JKO-type gradient flows with total-variation contraction. This approach unifies variational, lattice-based methods with OT theory and broadens the scope of robust, non-smooth comparison results in PDEs and gradient-flow contexts.
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.