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 distributions, 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最基本的SPDE之一——已被提出用于描述弱相互作用粒子系统中的密度涨落。其原始形式高度奇异，即使通过正则性结构与仿射控分布等方法也无法重整化，阻碍了其严格数学证明的实现。学界近期认识到引入适当正则化是自然的，例如通过形式空间离散化或截断高频噪声。本文证明，Dean-Kawasaki方程的离散化形式正则化确实能精确描述弱相互作用扩散粒子系统的密度涨落：我们证明在合适的弱度量下，离散化Dean-Kawasaki SPDE预测的涨落律逼近原始粒子系统的涨落律，误差为粒子数逆的任意高阶项与离散化误差。特别地，Dean-Kawasaki方程为实现弱相互作用粒子系统密度涨落的高效精确模拟提供了有效途径。