Using operator semigroup methods, we show that Fokker-Planck type second-order PDEs 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 $O(1/N^3)$.
"On the error of Fokker-Planck approximations of some one-step density dependent processes." Differential Integral Equations 33 (1/2) 67 - 90, January/February 2020.