{"paper":{"title":"Non-Uniqueness for Nonlinear Fokker--Planck Equations and Their Associated Distribution-Dependent SDEs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.PR","authors_text":"Huaxiang L\\\"u","submitted_at":"2026-06-30T11:20:46Z","abstract_excerpt":"In this paper, we study distribution-dependent stochastic differential equations on the domain $\\mathcal O=\\mathbb T^d$ or $\\mathbb R^d$, $d\\geq 2$, of the form \\begin{align*} {\\rm d}X_t = v(t,X_t,\\rho_t)\\,{\\rm d}t + \\sqrt{2}\\, \\sigma(t,X_t,\\rho_t)\\,{\\rm d}W_t, \\qquad \\rho_t:=\\frac{{\\rm d}\\mu_t}{{\\rm d}x}, \\end{align*} where $\\mu_t=\\operatorname{Law}(X_t)$. Our main construction is carried out at the level of the associated nonlinear Fokker--Planck equations. We first build non-unique probability solutions to these PDEs and then use the superposition principle to obtain non-unique martingale s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.31500","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.31500/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}