Dean–Kawasaki equation with initial condition in the space of positive distributions

We show that the Dean–Kawasaki equation does not admit non-trivial solutions in the space of tempered measures. More specifically, we consider martingale solutions taking values, and with initial conditions, in the subspace of measures admitting infinite mass and satisfying some integrability conditions. Following work by the first author, Lehmann and von Renesse (Electron Commun Probab 24:3916340, 2019), we show that the equation only admits solutions if the initial measure is a discrete measure. Our result extends the previously mentioned works by allowing measures with infinite mass.