Matrix displacement convexity along density flows

TL;DR

This paper introduces a stronger form of displacement convexity at the matrix level for density flows, enhancing classical inequalities and providing new insights into the behavior of these flows in various physical and mathematical models.

Contribution

It develops a novel matrix displacement convexity concept for density flows, enabling intrinsic dimensional analysis and improved inequalities in mean-field and related equations.

Findings

Matrix displacement convexity is stronger than classical notions.

Matrix differential inequalities facilitate dimensional functional inequalities.

Applications include turnpike properties and entropy bounds.

Abstract

A new notion of displacement convexity on a matrix level is developed for density flows arising from mean-field games, compressible Euler equations, entropic interpolation, and semi-classical limits of non-linear Schr\"odinger equations. Matrix displacement convexity is stronger than the classical notions of displacement convexity, and its verification (formal and rigorous) relies on matrix differential inequalities along the density flows. The matrical nature of these differential inequalities upgrades dimensional functional inequalities to their intrinsic dimensional counterparts, thus improving on many classical results. Applications include turnpike properties, evolution variational inequalities, and entropy growth bounds, which capture the behavior of the density flows along different directions in space.