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A gradient flow perspective on McKean-Vlasov equations in econophysics

David W. Cohen

TL;DR

This work develops a formal gradient-flow framework for a class of diffusion limits of kinetic asset exchange econophysics models by introducing a novel adapted metric ${\mathcal{C}_D}$ on a subspace of wealth distributions with fixed first moment. The Gini coefficient is shown to be a Lyapunov functional that drives the dynamics as a gradient flow within this geometry, providing a thermodynamics-like interpretation of inequality growth and a separation between energetics and kinetics via the geometry. The authors prove a monotonic increase of the Gini functional, demonstrate that the yard-sale model cannot be captured by classical $W_2$ gradient flows, and establish a suite of Sobolev-dual norm structures, a $ ho$-weighted biharmonic theory, and transport inequalities for the ${\mathcal{C}_D}$ metric. These results lay a formal foundation for understanding econophysically motivated dissipative dynamics and suggest new analytical and numerical directions beyond standard optimal transport theory.

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.

A gradient flow perspective on McKean-Vlasov equations in econophysics

TL;DR

This work develops a formal gradient-flow framework for a class of diffusion limits of kinetic asset exchange econophysics models by introducing a novel adapted metric on a subspace of wealth distributions with fixed first moment. The Gini coefficient is shown to be a Lyapunov functional that drives the dynamics as a gradient flow within this geometry, providing a thermodynamics-like interpretation of inequality growth and a separation between energetics and kinetics via the geometry. The authors prove a monotonic increase of the Gini functional, demonstrate that the yard-sale model cannot be captured by classical gradient flows, and establish a suite of Sobolev-dual norm structures, a -weighted biharmonic theory, and transport inequalities for the metric. These results lay a formal foundation for understanding econophysically motivated dissipative dynamics and suggest new analytical and numerical directions beyond standard optimal transport theory.

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.

Paper Structure

This paper contains 21 sections, 14 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Notation
  4. Kinetic asset exchange models and a thermodynamic second law
  5. Kinetic asset exchange models
  6. A thermodynamic second law for kinetic asset exchange
  7. An incompatibility result with the W2 gradient structure
  8. An adapted Wasserstein metric tensor
  9. Gradient flow formulation of kinetic asset exchange
  10. Conserved quantities of C D gradient flows
  11. The case of D[x,p]=p
  12. W2 and negative homogeneous Sobolev norms
  13. u-weighted biharmonic equation with Neumann boundary conditions
  14. The spaces H2u and H(-2)u
  15. Cp using the dual norm
  16. ...and 6 more sections

Key Result

Lemma 2.7

The second-order derivative of the Fréchet derivative of $G$ is $\rho(x)$.

Theorems & Definitions (36)

  • Definition 2.1: Wealth conserving
  • Definition 2.2: Positivity preserving
  • Definition 2.3: Unbiased exchange
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Proposition 2.8
  • proof
  • ...and 26 more