Energy distance and evolution problems: a promising tool for kinetic equations

TL;DR

The paper analyzes long-time convergence toward equilibrium for Fokker–Planck type equations with linear drift using Cramér and Energy distances. By deriving time-derivative identities for these distances in both one and multiple dimensions, it demonstrates exponential decay for drift-only and drift–diffusion FP equations, and identifies a polynomial decay rate of $t^{-(2-\alpha)/2}$ for the linear diffusion equation as a function of the distance order $\alpha$. Key contributions include explicit differential inequalities linking $E_\alpha$ and its negative-order counterpart, Fourier representations that connect to Sobolev norms, and applications to models with economic and social dynamics. The results offer sharper convergence benchmarks than entropy-based methods and suggest numerical advantages due to the distance’s tractable form via moments and Fourier terms.

Abstract

We study the rate of convergence to equilibrium of the solutions to Fokker-Planck type equations with linear drift by means of Cramér and Energy distances, which have been recently widely used in problems related to AI, in particular for tasks related to machine learning. In all cases in which the Fokker-Planck type equations can be treated through these distances, it is shown that the rate of decay is improved with respect to known results which are based on the decay of relative entropy.