Density fluctuations in weakly interacting particle systems via the Dean-Kawasaki equation

It has been proposed that the density fluctuations of a system of weakly interacting particles in the regime of large but finite particle number are captured by a highly singular SPDE -the Dean-Kawasaki equation. Due to the ever expanding use of the Dean-Kawasaki equation in applications, there is a desire to rigorously justify it. In this talk we show that, using a suitable weak distance, the law of the fluctuations as predicted by a spatially discretized Dean-Kawasaki equation approximates the law of the fluctuations of the particle system up to a term that is of arbitrary order in the inverse particle number and a numerical error. This talk is based on a joint work with Federico Cornalba, Julian Fischer, and Jonas Ingmanns.