article published yes Federico Cornalba author Julian L Fischer author 2C12A0B0-F248-11E8-B48F-1D18A9856A870000-0002-0479-558X Jonas Ingmanns author 71523d30-15b2-11ec-abd3-f80aa909d6b00009-0008-1310-7946 Claudia Raithel author JuFi department ISTplus - Postdoctoral Fellowships project Taming Complexity in Partial Differential Systems project 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. https://research-explorer.ista.ac.at/download/21894/21906/2026_AnnalsProbability_Cornalba.pdf application/pdfno Institute of Mathematical Statistics2026 eng Weakly interacting particle systemsfluctuating hydrodynamicsDean-Kawasaki equationstochastic PDEsnumerical approximation 0091-1798 2168-894X 2303.0042910.1214/25-aop1763 541155-215 Cornalba, Federico, Julian L Fischer, Jonas Ingmanns, and Claudia Raithel. “Density Fluctuations in Weakly Interacting Particle Systems via the Dean–Kawasaki Equation.” <i>The Annals of Probability</i>. Institute of Mathematical Statistics, 2026. <a href="https://doi.org/10.1214/25-aop1763">https://doi.org/10.1214/25-aop1763</a>. Cornalba, Federico, et al. “Density Fluctuations in Weakly Interacting Particle Systems via the Dean–Kawasaki Equation.” <i>The Annals of Probability</i>, vol. 54, no. 1, Institute of Mathematical Statistics, 2026, pp. 155–215, doi:<a href="https://doi.org/10.1214/25-aop1763">10.1214/25-aop1763</a>. F. Cornalba, J.L. Fischer, J. Ingmanns, C. Raithel, The Annals of Probability 54 (2026) 155–215. F. Cornalba, J. L. Fischer, J. Ingmanns, and C. Raithel, “Density fluctuations in weakly interacting particle systems via the Dean–Kawasaki equation,” <i>The Annals of Probability</i>, vol. 54, no. 1. Institute of Mathematical Statistics, pp. 155–215, 2026. Cornalba, F., Fischer, J. L., Ingmanns, J., &#38; Raithel, C. (2026). Density fluctuations in weakly interacting particle systems via the Dean–Kawasaki equation. <i>The Annals of Probability</i>. Institute of Mathematical Statistics. <a href="https://doi.org/10.1214/25-aop1763">https://doi.org/10.1214/25-aop1763</a> Cornalba F, Fischer JL, Ingmanns J, Raithel C. 2026. Density fluctuations in weakly interacting particle systems via the Dean–Kawasaki equation. The Annals of Probability. 54(1), 155–215. Cornalba F, Fischer JL, Ingmanns J, Raithel C. Density fluctuations in weakly interacting particle systems via the Dean–Kawasaki equation. <i>The Annals of Probability</i>. 2026;54(1):155-215. doi:<a href="https://doi.org/10.1214/25-aop1763">10.1214/25-aop1763</a> 218942026-05-20T08:25:25Z2026-05-21T07:21:25Z