{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2021:PEQ7EHVB5W3ZDPDK3QHDTH6CXQ","short_pith_number":"pith:PEQ7EHVB","schema_version":"1.0","canonical_sha256":"7921f21ea1edb791bc6adc0e399fc2bc1cf6b376fefb094f9e4a4125b693078f","source":{"kind":"arxiv","id":"2109.01273","version":2},"attestation_state":"computed","paper":{"title":"Second order McKean-Vlasov SDEs and kinetic Fokker-Planck-Kolmogorov equations","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.PR","authors_text":"Xicheng Zhang","submitted_at":"2021-09-03T02:14:26Z","abstract_excerpt":"In this paper we study second order stochastic differential equations with measurable and density-distribution dependent coefficients. Through establishing a maximum principle for kinetic Fokker-Planck-Kolmogorov equations with distribution-valued inhomogeneous term, we show the existence of weak solutions under mild assumptions. Moreover, by using the H\\\"older regularity estimate obtained recently in \\cite{GIMV19}, we also show the well-posedness of generalized martingale problems when diffusion coefficients only depend on the position variable (not necessarily continuous). Even in the non de"},"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":"2109.01273","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.PR","submitted_at":"2021-09-03T02:14:26Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"c77449b680bbb2c91a3ae36ad7b67f4fa18decf28d3a71101eb0fe1693747fc3","abstract_canon_sha256":"d68dbf38ae628ac69478d799ce4d56d82b64bf3ecd9f5cd3670241f551ee78d5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:51:09.428459Z","signature_b64":"NSK/VcVHX8u8fZJ6uWzexBb0J/bdb7LMvfMcS4JfXgR045mXE9VjTg2bMrAaelCm1ux5yxFAJdAWcQo8r0boCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7921f21ea1edb791bc6adc0e399fc2bc1cf6b376fefb094f9e4a4125b693078f","last_reissued_at":"2026-07-05T03:51:09.427962Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:51:09.427962Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Second order McKean-Vlasov SDEs and kinetic Fokker-Planck-Kolmogorov equations","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.PR","authors_text":"Xicheng Zhang","submitted_at":"2021-09-03T02:14:26Z","abstract_excerpt":"In this paper we study second order stochastic differential equations with measurable and density-distribution dependent coefficients. Through establishing a maximum principle for kinetic Fokker-Planck-Kolmogorov equations with distribution-valued inhomogeneous term, we show the existence of weak solutions under mild assumptions. Moreover, by using the H\\\"older regularity estimate obtained recently in \\cite{GIMV19}, we also show the well-posedness of generalized martingale problems when diffusion coefficients only depend on the position variable (not necessarily continuous). Even in the non de"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.01273","kind":"arxiv","version":2},"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/2109.01273/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2109.01273","created_at":"2026-07-05T03:51:09.428027+00:00"},{"alias_kind":"arxiv_version","alias_value":"2109.01273v2","created_at":"2026-07-05T03:51:09.428027+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.01273","created_at":"2026-07-05T03:51:09.428027+00:00"},{"alias_kind":"pith_short_12","alias_value":"PEQ7EHVB5W3Z","created_at":"2026-07-05T03:51:09.428027+00:00"},{"alias_kind":"pith_short_16","alias_value":"PEQ7EHVB5W3ZDPDK","created_at":"2026-07-05T03:51:09.428027+00:00"},{"alias_kind":"pith_short_8","alias_value":"PEQ7EHVB","created_at":"2026-07-05T03:51:09.428027+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2507.23553","citing_title":"McKean-Vlasov equations with singular coefficients - a review of recent results","ref_index":123,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/PEQ7EHVB5W3ZDPDK3QHDTH6CXQ","json":"https://pith.science/pith/PEQ7EHVB5W3ZDPDK3QHDTH6CXQ.json","graph_json":"https://pith.science/api/pith-number/PEQ7EHVB5W3ZDPDK3QHDTH6CXQ/graph.json","events_json":"https://pith.science/api/pith-number/PEQ7EHVB5W3ZDPDK3QHDTH6CXQ/events.json","paper":"https://pith.science/paper/PEQ7EHVB"},"agent_actions":{"view_html":"https://pith.science/pith/PEQ7EHVB5W3ZDPDK3QHDTH6CXQ","download_json":"https://pith.science/pith/PEQ7EHVB5W3ZDPDK3QHDTH6CXQ.json","view_paper":"https://pith.science/paper/PEQ7EHVB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2109.01273&json=true","fetch_graph":"https://pith.science/api/pith-number/PEQ7EHVB5W3ZDPDK3QHDTH6CXQ/graph.json","fetch_events":"https://pith.science/api/pith-number/PEQ7EHVB5W3ZDPDK3QHDTH6CXQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PEQ7EHVB5W3ZDPDK3QHDTH6CXQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PEQ7EHVB5W3ZDPDK3QHDTH6CXQ/action/storage_attestation","attest_author":"https://pith.science/pith/PEQ7EHVB5W3ZDPDK3QHDTH6CXQ/action/author_attestation","sign_citation":"https://pith.science/pith/PEQ7EHVB5W3ZDPDK3QHDTH6CXQ/action/citation_signature","submit_replication":"https://pith.science/pith/PEQ7EHVB5W3ZDPDK3QHDTH6CXQ/action/replication_record"}},"created_at":"2026-07-05T03:51:09.428027+00:00","updated_at":"2026-07-05T03:51:09.428027+00:00"}