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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2606.31500","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-06-30T11:20:46Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"c6b1bef44a81900e66ef3d50f4b28fad6d03fb2b474339a7ba41a77b6bde3a66","abstract_canon_sha256":"4d00bc81542202d8fd90748262b99730b3d0cfb280b589bae6c5f97c0620bc18"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-01T01:18:05.146879Z","signature_b64":"wxMtAtuv/8tcGJmRE9ZScGm2LdpcYtoHVMMUS4b48v8L2LPwKkZB3rZmcTP5EDEUnjanJ5WNxmNLfhIvJG2ZCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5d0385e3a649585f4df7b15efb285ed5bb4f506143f16fba712b67fc2457a4fa","last_reissued_at":"2026-07-01T01:18:05.146441Z","signature_status":"signed_v1","first_computed_at":"2026-07-01T01:18:05.146441Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\mu_t=\\operatorname{Law}(X_t)$. 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