The Dean-Kawasaki equation - one of the most fundamental SPDEs of fluctuating hydrodynamics - has been proposed as a model for density fluctuations in weakly interacting particle systems. In its original form it is highly singular and fails to be renormalizable even by approaches such as regularity structures and paracontrolled distrubutions, hindering mathematical approaches to its rigorous justification. It has been understood recently that it is natural to introduce a suitable regularization, e.g., by applying a formal spatial discretization or by truncating high-frequency noise. In the present work, we prove that a regularization in form of a formal discretization of the Dean-Kawasaki equation indeed accurately describes density fluctuations in systems of weakly interacting diffusing particles: We show that in suitable weak metrics, the law of fluctuations as predicted by the discretized Dean--Kawasaki SPDE approximates the law of fluctuations of the original particle system, up to an error that is of arbitrarily high order in the inverse particle number and a discretization error. In particular, the Dean-Kawasaki equation provides a means for efficient and accurate simulations of density fluctuations in weakly interacting particle systems.

翻译：Dean-Kawasaki方程——作为涨落流体动力学中最基础的SPDE之一——被提出用于描述弱相互作用粒子系统中的密度涨落。其原始形式具有高度奇异性，即便通过正则性结构与抛物控制分布等方法也无法实现重整化，这阻碍了对其严格化的数学研究。近期学界认识到，引入适当正则化（如形式空间离散化或截断高频噪声）是自然的处理方式。本文证明，Dean-Kawasaki方程的形式离散化正则形式确实能精确描述弱相互作用扩散粒子系统中的密度涨落：在合适的弱度量下，离散化Dean-Kawasaki SPDE所预测的涨落律收敛于原始粒子系统的涨落律，其误差包含任意高阶的粒子逆幂次项与离散化误差。特别地，该方程为弱相互作用粒子系统中密度涨落的高效精确模拟提供了有效途径。