@article{21894,
  abstract     = {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, for example, by applying a formal spatial discretization or by truncating high-frequency noise: This yields
well-posed equations that should still precisely approximate the law of the
particle density fluctuations.
In the present work, we prove that a regularization in the 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.},
  author       = {Cornalba, Federico and Fischer, Julian L and Ingmanns, Jonas and Raithel, Claudia},
  issn         = {2168-894X},
  journal      = {The Annals of Probability},
  keywords     = {Weakly interacting particle systems, fluctuating hydrodynamics, Dean-Kawasaki equation, stochastic PDEs, numerical approximation},
  number       = {1},
  pages        = {155--215},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Density fluctuations in weakly interacting particle systems via the Dean–Kawasaki equation}},
  doi          = {10.1214/25-aop1763},
  volume       = {54},
  year         = {2026},
}

@article{10613,
  abstract     = {Motivated by the recent preprint [\emph{arXiv:2004.08412}] by Ayala, Carinci, and Redig, we first provide a general framework for the study of scaling limits of higher-order fields. Then, by considering the same class of infinite interacting particle systems as in [\emph{arXiv:2004.08412}], namely symmetric simple exclusion and inclusion processes in the d-dimensional Euclidean lattice, we prove the hydrodynamic limit, and convergence for the equilibrium fluctuations, of higher-order fields. In particular, the limit fields exhibit a tensor structure. Our fluctuation result differs from that in [\emph{arXiv:2004.08412}], since we considered-dimensional Euclidean lattice, we prove the hydrodynamic limit, and convergence for the equilibrium fluctuations, of higher-order fields. In particular, the limit fields exhibit a tensor structure. Our fluctuation result differs from that in [\emph{arXiv:2004.08412}], since we consider a different notion of higher-order fluctuation fields.},
  author       = {Chen, Joe P. and Sau, Federico},
  issn         = {1024-2953},
  journal      = {Markov Processes And Related Fields},
  keywords     = {interacting particle systems, higher-order fields, hydrodynamic limit, equilibrium fluctuations, duality},
  number       = {3},
  pages        = {339--380},
  publisher    = {Polymat Publishing},
  title        = {{Higher-order hydrodynamics and equilibrium fluctuations of interacting particle systems}},
  volume       = {27},
  year         = {2021},
}