2024/09/23 by Fei Cao, Cao, Fei · 1 citation
Physics and Astronomy · #34A35 #60J28 #91B70 #91B80 #Advanced Thermodynamics and Statistical Mechanics #Classical Analysis and ODEs (math.CA) #FOS: Economics and business #FOS: Mathematics #Mathematical Finance (q-fin.MF) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2409.15225
openalex publication_date 2024/09/23 · openalex created_date 2024/10/26 · openalex updated_date 2026/07/28
In several recent works on infinite-dimensional systems of ODEs \citecaoderivation2021,caoexplicit2021,caoiterative2024,caosticky2024, which arise from the mean-field limit of agent-based models in economics and social sciences and model the evolution of probability distributions (on the set of non-negative integers), it is often shown that the Gini index serves as a natural Lyapunov functional along the solution to a given system. Furthermore, the Gini index converges to that of the equilibrium distribution. However, it is not immediately clear whether this convergence at the level of the Gini index implies convergence in the sense of probability distributions or even stronger notions of convergence. In this paper, we prove several results in this direction, highlighting the interplay between the Gini index and other popular metrics, such as the Wasserstein distance and the usual ℓp distance, which are used to quantify the closeness of probability distributions.