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

Abstract

We prove that the Dean-Kawasaki-type stochastic partial differential equation $$\partial \rho =\nabla \cdot (\sqrt{\rho }\xi )+\nabla \cdot \left(\rho H(\rho )\right)$$ 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 ($H\equiv 0$). The result is sharp, in the sense that solutions are known to exist for some unbounded $H$.

9 pages

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