# Variational problems involving unequal dimensional optimal transport

## TL;DR

This paper investigates variational problems involving optimal transport between unequal dimensional spaces, establishing conditions for tractability, deriving differential equations, and proving convergence and regularity of solutions.

## Contribution

It introduces a nestedness condition for minimizers in unequal dimensional optimal transport problems and develops methods to analyze and compute solutions.

## Key findings

- Nestedness condition holds for minimizers, improving problem tractability.
- Derived local differential equations characterizing solutions.
- Proved convergence of an iterative scheme and regularity of solutions.

## Abstract

This paper is devoted to variational problems on the set of probability measures which involve optimal transport between unequal dimensional spaces. In particular, we study the minimization of a functional consisting of the sum of a term reflecting the cost of (unequal dimensional) optimal transport between one fixed and one free marginal, and another functional of the free marginal (of various forms). Motivating applications include Cournot-Nash equilibria where the strategy space is lower dimensional than the space of agent types. For a variety of different forms of the term described above, we show that a nestedness condition, which is known to yield much improved tractability of the optimal transport problem, holds for any minimizer. Depending on the exact form of the functional, we exploit this to find local differential equations characterizing solutions, prove convergence of an iterative scheme to compute the solution, and prove regularity results.