{"paper":{"title":"Dean-Kawasaki Equation with Biot-Savart and Keller-Segel Interactions: Existence and Large Deviations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The Dean-Kawasaki equation admits probabilistically weak renormalized kinetic solutions even with singular Biot-Savart and Keller-Segel interactions.","cross_cats":["math.AP"],"primary_cat":"math.PR","authors_text":"Xiaohao Ji, Yue Sun, Zhengyan Wu","submitted_at":"2026-05-13T13:05:51Z","abstract_excerpt":"We establish the existence of probabilistically weak, renormalized kinetic solutions to the Dean--Kawasaki equation with singular interaction kernels, including those of Biot--Savart and Keller--Segel type. Under a suitable regularization of the square-root noise coefficient, we further prove a restricted large deviation principle for probabilistically weak solutions to the regularized Dean--Kawasaki equation. The Biot--Savart and Keller--Segel type interactions introduce a scaling criticality within the $L^1$ framework of the Dean--Kawasaki equation and the associated skeleton equation, which"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We establish the existence of probabilistically weak, renormalized kinetic solutions to the Dean--Kawasaki equation with singular interaction kernels, including those of Biot--Savart and Keller--Segel type. Under a suitable regularization of the square-root noise coefficient, we further prove a restricted large deviation principle for probabilistically weak solutions to the regularized Dean--Kawasaki equation.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"A suitable regularization of the square-root noise coefficient exists that preserves the Dean-Kawasaki structure while allowing the exponential tightness argument and weak-strong uniqueness to close the large-deviation proof despite the scaling criticality of the singular kernels.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Existence of probabilistically weak renormalized kinetic solutions and a restricted large deviation principle are established for the Dean-Kawasaki equation with Biot-Savart and Keller-Segel singular kernels via regularization and a new exponential tightness argument.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The Dean-Kawasaki equation admits probabilistically weak renormalized kinetic solutions even with singular Biot-Savart and Keller-Segel interactions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a669e0cbcdcc1ae183512e305076cf90f4486562539b0a87b3d63fc5cc42a702"},"source":{"id":"2605.13479","kind":"arxiv","version":1},"verdict":{"id":"c3887170-c05a-4531-8c5c-2a2e64c3406f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:30:14.818807Z","strongest_claim":"We establish the existence of probabilistically weak, renormalized kinetic solutions to the Dean--Kawasaki equation with singular interaction kernels, including those of Biot--Savart and Keller--Segel type. Under a suitable regularization of the square-root noise coefficient, we further prove a restricted large deviation principle for probabilistically weak solutions to the regularized Dean--Kawasaki equation.","one_line_summary":"Existence of probabilistically weak renormalized kinetic solutions and a restricted large deviation principle are established for the Dean-Kawasaki equation with Biot-Savart and Keller-Segel singular kernels via regularization and a new exponential tightness argument.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"A suitable regularization of the square-root noise coefficient exists that preserves the Dean-Kawasaki structure while allowing the exponential tightness argument and weak-strong uniqueness to close the large-deviation proof despite the scaling criticality of the singular kernels.","pith_extraction_headline":"The Dean-Kawasaki equation admits probabilistically weak renormalized kinetic solutions even with singular Biot-Savart and Keller-Segel interactions."},"references":{"count":13,"sample":[{"doi":"10.1214/10-aihp382","year":2011,"title":"Variational representations for continuous time processes.Ann","work_id":"1feeca22-324e-4692-a1cd-1939db419328","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1016/j.crma.2019.09.007","year":2019,"title":"[CD16] Jurandir Ceccon and Carlos E","work_id":"42f4a61b-f354-482c-860a-44b898d10288","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1002/9781118165904","year":1996,"title":"[Dea96] David S","work_id":"f19f0369-05af-4b32-97b4-cb915d049381","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1214/20-ejp436","year":2026,"title":"Conservative stochastic PDE and fluctuations of the sym- metric simple exclusion process.Communications in Mathematical Physics, 407:Paper No","work_id":"2a52fc70-31a6-411b-b797-b49fa7927ac8","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1007/s40072-024-00324-1","year":null,"title":"[Due16] Mitia Duerinckx","work_id":"f6bcf05e-0bb6-4b14-b9c7-aa3f2302934a","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":13,"snapshot_sha256":"6d83c02abe89fe6e0186f0fdff33eefe28b1f73928bb76a36a492847678a514d","internal_anchors":2},"formal_canon":{"evidence_count":2,"snapshot_sha256":"602f667598ba5fb9528793d2fc96ed483f661d21da828926b8f679bb220ad5ae"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}