A Dynamical Equation for the Lorenz Curve: Dynamics of incomplete moments of probability distributions arising from Fokker-Planck equations

TL;DR

This paper introduces a new dynamical framework for analyzing the evolution of probability distributions, especially wealth distributions, using incomplete moments derived from Fokker-Planck equations, simplifying the analysis over a compact domain.

Contribution

It develops a novel approach transforming Fokker-Planck equations into Lorenz dynamics over incomplete moments, enabling easier analysis of distribution evolution.

Findings

Derived Lorenz dynamics for heat and kinetic equations

Applied the framework to economic models of wealth distribution

Provided solutions for specific Fokker-Planck variants

Abstract

Fokker-Planck equations (forward Kolmogorov equations) evolve probability densities in time from an initial condition. For distributions over the real line, these evolution equations can sometimes be transformed into dynamics over the incomplete zeroth and first moments. We call this perspective the Lorenz dynamics of the system after the Lorenz curve description of distributions of wealth. This offers the benefit of presenting the dynamics over a compact domain. The integral transformation is motivated and then stated for a general class of Fokker-Planck equations. Following this, the transformed equation is solved for the heat equation and some variants thereof. Finally, some equations arising from the application of kinetic theory to idealized economic systems are transformed and analyzed in this new light.