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

TL;DR

This paper proves that the pure-noise Dean-Kawasaki equation, a stochastic PDE modeling particle densities, has no solutions for any initial conditions, highlighting fundamental ill-posedness issues in such stochastic models.

Contribution

It establishes the non-existence of solutions for the Dean-Kawasaki equation with bounded measurable functions H, including the pure-noise case, clarifying the limits of well-posedness.

Findings

No solutions exist for the pure-noise Dean-Kawasaki equation.

Solutions exist only for certain unbounded functions H.

The result is sharp, indicating the boundary of well-posedness.

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 $H$.