A gradient flow perspective on McKean-Vlasov equations in econophysics

TL;DR

This paper establishes a gradient flow framework for McKean-Vlasov equations in econophysics, using the Gini coefficient as a Lyapunov functional and introducing a novel Riemannian geometry on probability densities.

Contribution

It introduces a new geometric perspective on econophysical equations by linking the Gini coefficient to gradient flows and deriving associated transport inequalities.

Findings

Gini coefficient acts as a Lyapunov functional for certain equations.

A new Riemannian metric on probability densities is constructed.

Transport inequalities related to the new metric are proven.

Abstract

We prove that the Gini coefficient of economic inequality is a Lyapunov functional for a class of nonlinear, nonlocal integro-differential equations arising at the intersection of mathematics, economics, and statistical physics. Next, a novel Riemannian geometry is imposed on a subset of probability densities such that the evolutionary dynamics are formally driven by the Gini coefficient functional as a gradient flow. Thus in the same way that classical 2-Wasserstein theory connects heat flow and the Second Law of Thermodynamics by way of Boltzmann entropy, the work here gives rise to a principle of econophysics that is much of the same flavor but for the Gini coefficient. The noncanonical Onsager operators associated to the metric tensors are derived and some transport inequalities proven. The new metric relates to the dual norm of a second-order Sobolev-like factor space, in a similar way to how the classical 2-Wasserstein metric linearizes as the dual norm of a first-order, homogeneous Sobolev space.