[{"ec_funded":1,"status":"public","quality_controlled":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"article_type":"original","intvolume":"        54","title":"Density fluctuations in weakly interacting particle systems via the Dean–Kawasaki equation","arxiv":1,"day":"01","keyword":["Weakly interacting particle systems","fluctuating hydrodynamics","Dean-Kawasaki equation","stochastic PDEs","numerical approximation"],"date_updated":"2026-05-21T07:21:25Z","date_created":"2026-05-20T08:25:25Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"publisher":"Institute of Mathematical Statistics","fulldoi":"https://doi.org/10.1214/25-aop1763","month":"01","PlanS_conform":"1","ddc":["510"],"_id":"21894","article_processing_charge":"Yes (in subscription journal)","APC_amount":"1352,08 EUR","scopus_import":"1","department":[{"_id":"JuFi"}],"acknowledgement":"All authors gratefully acknowledge funding from the Austrian Science Fund (FWF) through the project F65. CR gratefully acknowledges support from the Austrian Science Fund (FWF), grants P30000, P33010, W1245. FC gratefully acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411.","OA_type":"hybrid","type":"journal_article","has_accepted_license":"1","publication_status":"published","oa_version":"Published Version","OA_place":"publisher","doi":"10.1214/25-aop1763","year":"2026","volume":54,"citation":{"ama":"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>","ista":"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.","ieee":"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.","short":"F. Cornalba, J.L. Fischer, J. Ingmanns, C. Raithel, The Annals of Probability 54 (2026) 155–215.","mla":"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>.","chicago":"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>.","apa":"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>"},"file_date_updated":"2026-05-21T07:11:27Z","corr_author":"1","author":[{"last_name":"Cornalba","first_name":"Federico","full_name":"Cornalba, Federico"},{"orcid":"0000-0002-0479-558X","last_name":"Fischer","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87","first_name":"Julian L","full_name":"Fischer, Julian L"},{"orcid":"0009-0008-1310-7946","last_name":"Ingmanns","first_name":"Jonas","id":"71523d30-15b2-11ec-abd3-f80aa909d6b0","full_name":"Ingmanns, Jonas"},{"first_name":"Claudia","last_name":"Raithel","full_name":"Raithel, Claudia"}],"issue":"1","page":"155-215","date_published":"2026-01-01T00:00:00Z","language":[{"iso":"eng"}],"file":[{"date_created":"2026-05-21T07:11:27Z","date_updated":"2026-05-21T07:11:27Z","file_name":"2026_AnnalsProbability_Cornalba.pdf","checksum":"3e60c0e25a1c96342029a7d2b031505f","file_id":"21906","relation":"main_file","access_level":"open_access","file_size":865745,"content_type":"application/pdf","creator":"dernst","success":1}],"abstract":[{"text":"The Dean–Kawasaki equation—one of the most fundamental SPDEs of\r\nfluctuating hydrodynamics—has been proposed as a model for density fluctuations in weakly interacting particle systems. In its original form, it is highly\r\nsingular 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\r\nnatural to introduce a suitable regularization, for example, by applying a formal spatial discretization or by truncating high-frequency noise: This yields\r\nwell-posed equations that should still precisely approximate the law of the\r\nparticle density fluctuations.\r\nIn the present work, we prove that a regularization in the form of a formal\r\ndiscretization of the Dean–Kawasaki equation indeed accurately describes\r\ndensity fluctuations in systems of weakly interacting diffusing particles: We\r\nshow that, in suitable weak metrics, the law of fluctuations as predicted by\r\nthe discretized Dean–Kawasaki SPDE approximates the law of fluctuations\r\nof the original particle system, up to an error that is of arbitrarily high order in\r\nthe inverse particle number and a discretization error. In particular, the Dean–\r\nKawasaki equation provides a means for efficient and accurate simulations of\r\ndensity fluctuations in weakly interacting particle systems.","lang":"eng"}],"project":[{"grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships","call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425"},{"_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","grant_number":"F6504","name":"Taming Complexity in Partial Differential Systems"}],"publication_identifier":{"issn":["0091-1798"],"eissn":["2168-894X"]},"publication":"The Annals of Probability","external_id":{"arxiv":["2303.00429"]}},{"author":[{"full_name":"Chen, Joe P.","last_name":"Chen","first_name":"Joe P."},{"full_name":"Sau, Federico","id":"E1836206-9F16-11E9-8814-AEFDE5697425","last_name":"Sau","first_name":"Federico"}],"page":"339-380","issue":"3","project":[{"_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships"}],"publication_identifier":{"issn":["1024-2953"]},"external_id":{"arxiv":["2008.13403"]},"publication":"Markov Processes And Related Fields","main_file_link":[{"url":"https://arxiv.org/abs/2008.13403","open_access":"1"}],"language":[{"iso":"eng"}],"date_published":"2021-03-16T00:00:00Z","abstract":[{"text":"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.","lang":"eng"}],"publication_status":"published","oa_version":"Preprint","department":[{"_id":"JaMa"}],"type":"journal_article","acknowledgement":"F.S. would like to thank Mario Ayala and Frank Redig for useful discussions. J.P.C. acknowledges partial financial support from the US National Science Foundation (DMS-1855604). F.S. was financially supported by the European Union’s Horizon 2020 research and innovation programme under the Marie-Skłodowska-Curie grant agreement No. 754411.\r\n","citation":{"ieee":"J. P. Chen and F. Sau, “Higher-order hydrodynamics and equilibrium fluctuations of interacting particle systems,” <i>Markov Processes And Related Fields</i>, vol. 27, no. 3. Polymat Publishing, pp. 339–380, 2021.","ista":"Chen JP, Sau F. 2021. Higher-order hydrodynamics and equilibrium fluctuations of interacting particle systems. Markov Processes And Related Fields. 27(3), 339–380.","ama":"Chen JP, Sau F. Higher-order hydrodynamics and equilibrium fluctuations of interacting particle systems. <i>Markov Processes And Related Fields</i>. 2021;27(3):339-380.","chicago":"Chen, Joe P., and Federico Sau. “Higher-Order Hydrodynamics and Equilibrium Fluctuations of Interacting Particle Systems.” <i>Markov Processes And Related Fields</i>. Polymat Publishing, 2021.","apa":"Chen, J. P., &#38; Sau, F. (2021). Higher-order hydrodynamics and equilibrium fluctuations of interacting particle systems. <i>Markov Processes And Related Fields</i>. Polymat Publishing.","mla":"Chen, Joe P., and Federico Sau. “Higher-Order Hydrodynamics and Equilibrium Fluctuations of Interacting Particle Systems.” <i>Markov Processes And Related Fields</i>, vol. 27, no. 3, Polymat Publishing, 2021, pp. 339–80.","short":"J.P. Chen, F. Sau, Markov Processes And Related Fields 27 (2021) 339–380."},"volume":27,"related_material":{"link":[{"url":"http://math-mprf.org/journal/articles/id1614/","description":"Link to Abstract on publisher's website","relation":"other"},{"description":"Referred to in Abstract","url":"https://arxiv.org/abs/2004.08412","relation":"used_for_analysis_in"}]},"corr_author":"1","year":"2021","oa":1,"date_updated":"2025-06-26T11:57:53Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2022-01-10T14:02:31Z","keyword":["interacting particle systems","higher-order fields","hydrodynamic limit","equilibrium fluctuations","duality"],"_id":"10613","article_processing_charge":"No","month":"03","publisher":"Polymat Publishing","quality_controlled":"1","article_type":"original","status":"public","ec_funded":1,"day":"16","title":"Higher-order hydrodynamics and equilibrium fluctuations of interacting particle systems","intvolume":"        27","arxiv":1}]