Quantitative convergence guarantees for the mean-field dispersion process

TL;DR

This paper provides quantitative convergence guarantees for the mean-field dispersion process by analyzing a nonlinear integral equation, with results supported by numerical simulations across different particle regimes.

Contribution

It introduces a novel analysis of a nonlinear Volterra-type integral equation to establish convergence rates for the mean-field dispersion process.

Findings

Convergence guarantees depend on the regime of particle number per site.

Numerical simulations confirm theoretical long-time behavior.

Analysis applies to various regimes of the mean-field dynamics.

Abstract

We study the discrete Fokker-Planck equation associated with the mean-field dynamics of a particle system called the dispersion process. For different regimes of the average number of particles per site (denoted by $\mu > 0$), we establish various quantitative long-time convergence guarantees toward the global equilibrium (depending on the sign of $\mu - 1$), which is also confirmed by numerical simulations. The main novelty/contribution of this manuscript lies in the careful and tricky analysis of a nonlinear Volterra-type integral equation satisfied by a key auxiliary function.