Quantitative convergence guarantees for the mean-field dispersion process

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 $μ> 0$), we establish various quantitative long-time convergence guarantees toward the global equilibrium (depending on the sign of $μ- 1$), 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.