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

We prove that the Dean–Kawasaki-type stochastic partial differential equation ∂μ=∇⋅(μξ)+∇⋅(μH(μ)), with vector-valued space-time white noise ξ, 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≡0). The result is sharp, in the sense that solutions are known to exist for some unbounded H.