A regularised Dean-Kawasaki model: derivation and analysis

TL;DR

This paper derives and analyzes a regularised Dean-Kawasaki model of wave equation type driven by coloured noise, establishing existence and uniqueness of solutions for a finite-sized particle system in one dimension.

Contribution

It introduces a regularised Dean-Kawasaki model for second order Langevin dynamics and proves existence and uniqueness of solutions, addressing open well-posedness issues.

Findings

Established existence and uniqueness of solutions for the regularised model

Demonstrated the model's well-posedness with high probability

Interpreted regularisation as considering finite-sized particles

Abstract

The Dean-Kawasaki model consists of a nonlinear stochastic partial differential equation featuring a conservative, multiplicative, stochastic term with non-Lipschitz coefficient, and driven by space-time white noise; this equation describes the evolution of the density function for a system of finitely many particles governed by Langevin dynamics. Well-posedness for the Dean-Kawasaki model is open except for specific diffusive cases, corresponding to overdamped Langevin dynamics. There, it was recently shown by Lehmann, Konarovskyi, and von Renesse that no regular (non-atomic) solutions exist. We derive and analyse a suitably regularised Dean-Kawasaki model of wave equation type driven by coloured noise, corresponding to second order Langevin dynamics, in one space dimension. The regularisation can be interpreted as considering particles of finite size rather than describing them by atomic measures. We establish existence and uniqueness of a solution. Specifically, we prove a high-probability result for the existence and uniqueness of mild solutions to this regularised Dean-Kawasaki model.