Dynamical equation for the Lorenz curve: Dynamics of incomplete moments of probability distributions arising from Fokker-Planck equations.

Abstract

Fokker-Planck equations (forward Kolmogorov equations) describe the time evolution of probability densities from a given 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.