Ill-posedness vs. triviality for Dean-Kawasaki dynamics with smooth interaction

The Dean-Kawasaki equation is an SPDE, which is used most notably in non-equilibrium statistical mechanics in order to model the evolution of fluctuating density fields. Apart from the physical reasoning, it appears as a natural candidate for a “Wasserstein-Diffusion”. However, we will prove that such dynamics are degenerate, in that the equation admits either only atomic or no solutions at all. This is joint work with Vitalii Konarovskyi and Max von Renesse.