Weak Error of Dean-Kawasaki Equation with Smooth Mean-Field Interactions

TL;DR

This paper analyzes the weak-error rate of a regularized Dean-Kawasaki SPDE with smooth mean-field interactions, establishing existence, uniqueness, and error estimates for particle systems with noise.

Contribution

It provides the first weak-error rate analysis for the Dean-Kawasaki equation with smooth mean-field interactions, extending previous results to more general settings.

Findings

Weak-error rate matches previous free Brownian particle results.

Global existence and uniqueness of the SPDEs are proven.

Error estimates are obtained using Kolmogorov equations on probability measures.

Abstract

We consider the weak-error rate of the SPDE approximation by regularized Dean-Kawasaki equation with It\^o noise for particle systems with mean-field interactions both on the drift and the noise. The global existence and uniqueness of the corresponding SPDEs are established using the variational approach to SPDEs, and the weak-error rate is estimated using the technique of Kolmogorov equations on the space of probability measures. In particular, the rate derived in this paper coincides with that is the previous work arXiv:2212.11714, which considered free Brownian particles using Laplace duality.