On the error of Fokker-Planck approximations of some one-step density dependent processes

TL;DR

This paper demonstrates that Fokker-Planck PDEs can effectively approximate the distribution evolution of large particle systems, providing error bounds of order 1/N and potential for even tighter bounds.

Contribution

The paper introduces operator semigroup methods to establish error bounds for Fokker-Planck approximations of one-step density-dependent processes, improving upon previous mean-field results.

Findings

Error bound of order O(1/N) for Fokker-Planck approximations

Potential to achieve error bounds up to O(1/N^3)

Method surpasses previous mean-field approximation accuracy

Abstract

Using operator semigroup methods, we show that Fokker-Planck type second-order PDE-s can be used to approximate the evolution of the distribution of a one-step process on $N$ particles governed by a large system of ODEs. The error bound is shown to be of order $O(1/N)$, surpassing earlier results that yielded this order for the error only for the expected value of the process, through mean-field approximations. We also present some conjectures showing that the methods used have the potential to yield even stronger bounds, up to $O(1/N^3)$.