Variational principles on the space of Lorenz curves: Gradient structures and isometries inspired by Wasserstein geometry

TL;DR

This paper develops new Riemannian gradient structures on Lorenz curves inspired by Wasserstein geometry, establishing isometries and variational principles that connect probability measures with nonlinear evolution equations.

Contribution

It introduces novel geometric structures and isometry results on Lorenz curves, linking them to Wasserstein geometry and nonlinear PDEs derived from Fokker-Planck equations.

Findings

Established Riemannian gradient structures on Lorenz curves.

Proved isometry results between probability measures and Lorenz curves.

Derived variational principles for nonlinear integro-differential equations.

Abstract

We motivate and derive novel Riemannian gradient structures on the space of Lorenz curves, which preserve infinite-dimensional variational principles inherited from Fokker-Planck equations via the lens of Wasserstein geometry and its variants. We also prove isometry results between corresponding formal manifolds of probability measures and Lorenz curves, which suggest meaningful metrics on the space of Lorenz curves when an underlying kinetic premise is present. In so doing, elegant variational principles are imbued upon highly nonlinear and nonlocal integro-differential evolution equations resulting from a recently derived variable transformation of McKean-Vlasov Fokker-Planck equations.