Ill-posedness of the pure-noise Dean–Kawasaki equation

We prove that the Dean-Kawasaki-type stochastic partial differential equation partial derivative mu = del (root mu xi) + del (mu H(mu)) , with vector-valued space-time white noise xi, does not admit solutions for any initial measure and any vector-valued bounded measurable function H on the space of measures. This applies in particular to the pure-noise Dean-Kawasaki equation (H equivalent to 0). The result is sharp, in the sense that solutions are known to exist for some unbounded H.