Regularisation and analysis of the Dean-Kawasaki model

The Dean-Kawasaki (DK) model is an important nonlinear stochastic equation in the fluctuating hydrodynamics field. It describes the evolution of the density function for a system of finitely many particles undergoing Langevin dynamics.

This equation is formally obtained, in a Schwartz distribution setting, on the hydrodynamic scale: due to its complicated noise structure, well-posedness for the equation is open except for the simplest, purely diffusive, case, corresponding to overdamped Langevin dynamics. In this case it was recently shown by Lehmann, Konarovskyi, and von Renesse that no regular (non-atomic) solutions exist.

We derive and analyse a suitably regularised DK model under two relevant sets of assumptions concerning the Langevin particle system. We thus address formal mathematical issues associated with the distributional setting of the original DK model.

We further prove a high-probability result for the existence and uniqueness of mild solutions to this regularised DK model, and also discuss relevant open questions.

This is joint work with Tony Shardlow and Johannes Zimmer.