{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:EUGSUEABBY7OZUSQBBUY5BIIQZ","short_pith_number":"pith:EUGSUEAB","canonical_record":{"source":{"id":"2501.10987","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.PR","submitted_at":"2025-01-19T08:47:32Z","cross_cats_sorted":[],"title_canon_sha256":"f56fa0050bf87988ebf1de2b98514cfa86d33c5d0807f9336c9e2dd44d8ca878","abstract_canon_sha256":"6995c8cbd1f8e848732301104c8ca08ecce39d06a763ae01ecf2b247dee306a2"},"schema_version":"1.0"},"canonical_sha256":"250d2a10010e3eecd25008698e8508867acbba4991dd995a40017cf6cd564367","source":{"kind":"arxiv","id":"2501.10987","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2501.10987","created_at":"2026-07-05T10:41:19Z"},{"alias_kind":"arxiv_version","alias_value":"2501.10987v4","created_at":"2026-07-05T10:41:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2501.10987","created_at":"2026-07-05T10:41:19Z"},{"alias_kind":"pith_short_12","alias_value":"EUGSUEABBY7O","created_at":"2026-07-05T10:41:19Z"},{"alias_kind":"pith_short_16","alias_value":"EUGSUEABBY7OZUSQ","created_at":"2026-07-05T10:41:19Z"},{"alias_kind":"pith_short_8","alias_value":"EUGSUEAB","created_at":"2026-07-05T10:41:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:EUGSUEABBY7OZUSQBBUY5BIIQZ","target":"record","payload":{"canonical_record":{"source":{"id":"2501.10987","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.PR","submitted_at":"2025-01-19T08:47:32Z","cross_cats_sorted":[],"title_canon_sha256":"f56fa0050bf87988ebf1de2b98514cfa86d33c5d0807f9336c9e2dd44d8ca878","abstract_canon_sha256":"6995c8cbd1f8e848732301104c8ca08ecce39d06a763ae01ecf2b247dee306a2"},"schema_version":"1.0"},"canonical_sha256":"250d2a10010e3eecd25008698e8508867acbba4991dd995a40017cf6cd564367","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:41:19.747851Z","signature_b64":"F3YFv0s0XgseAq9WHhx2TR3OWiHlaX2exsdUecFopSbAsylLy590mYdh92IeykrchwxCW+eLUbwFXEi6wNXsBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"250d2a10010e3eecd25008698e8508867acbba4991dd995a40017cf6cd564367","last_reissued_at":"2026-07-05T10:41:19.747282Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:41:19.747282Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2501.10987","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T10:41:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9G0q1Qsk/zqfnFdbLxJ3WdHFO1naNiSEdANr9RbpY/xC+J7BnoRYN54GRQKUB4F2VDyytxDe6Ua3pc+8aPzxBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T12:36:44.135609Z"},"content_sha256":"fade5e11a7b8259bfe4618b1a3f8aca980dff325d398d9f105d656a0e54ad357","schema_version":"1.0","event_id":"sha256:fade5e11a7b8259bfe4618b1a3f8aca980dff325d398d9f105d656a0e54ad357"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:EUGSUEABBY7OZUSQBBUY5BIIQZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Well-posedness of kinetic McKean-Vlasov equations","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alessio Rondelli, Andrea Pascucci","submitted_at":"2025-01-19T08:47:32Z","abstract_excerpt":"We consider the McKean-Vlasov equation $dX_t = b(t, X_t, [X_t])dt + \\sigma(t, X_t, [X_t])dW_t$ where $[X_t]$ is the law of $X_t$. We specifically consider the kinetic case, where the equation is degenerate because the dimension of the Brownian motion $W$ is strictly smaller than that of the solution $X$, as commonly required in classical models of collisional kinetic theory. Assuming H\\\"older continuous coefficients and a weak H\\\"ormander condition, we prove the well-posedness of the equation. This result advances the existing literature by filling a crucial gap: it addresses the previously un"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.10987","kind":"arxiv","version":4},"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/2501.10987/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T10:41:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eP86+YmN+2CL5qgvAp4xr5xBirAQNJ5hPGYjtu6aR6zN5jUkiSH/OJdb9iJPaBwjpgGVWEeR1ECOYtxQXvIwBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T12:36:44.136448Z"},"content_sha256":"607ee48372b19cae10eea7b835b745f83852b925005fea56abb69d73607eefba","schema_version":"1.0","event_id":"sha256:607ee48372b19cae10eea7b835b745f83852b925005fea56abb69d73607eefba"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EUGSUEABBY7OZUSQBBUY5BIIQZ/bundle.json","state_url":"https://pith.science/pith/EUGSUEABBY7OZUSQBBUY5BIIQZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EUGSUEABBY7OZUSQBBUY5BIIQZ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-15T12:36:44Z","links":{"resolver":"https://pith.science/pith/EUGSUEABBY7OZUSQBBUY5BIIQZ","bundle":"https://pith.science/pith/EUGSUEABBY7OZUSQBBUY5BIIQZ/bundle.json","state":"https://pith.science/pith/EUGSUEABBY7OZUSQBBUY5BIIQZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EUGSUEABBY7OZUSQBBUY5BIIQZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:EUGSUEABBY7OZUSQBBUY5BIIQZ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"6995c8cbd1f8e848732301104c8ca08ecce39d06a763ae01ecf2b247dee306a2","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.PR","submitted_at":"2025-01-19T08:47:32Z","title_canon_sha256":"f56fa0050bf87988ebf1de2b98514cfa86d33c5d0807f9336c9e2dd44d8ca878"},"schema_version":"1.0","source":{"id":"2501.10987","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2501.10987","created_at":"2026-07-05T10:41:19Z"},{"alias_kind":"arxiv_version","alias_value":"2501.10987v4","created_at":"2026-07-05T10:41:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2501.10987","created_at":"2026-07-05T10:41:19Z"},{"alias_kind":"pith_short_12","alias_value":"EUGSUEABBY7O","created_at":"2026-07-05T10:41:19Z"},{"alias_kind":"pith_short_16","alias_value":"EUGSUEABBY7OZUSQ","created_at":"2026-07-05T10:41:19Z"},{"alias_kind":"pith_short_8","alias_value":"EUGSUEAB","created_at":"2026-07-05T10:41:19Z"}],"graph_snapshots":[{"event_id":"sha256:607ee48372b19cae10eea7b835b745f83852b925005fea56abb69d73607eefba","target":"graph","created_at":"2026-07-05T10:41:19Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2501.10987/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider the McKean-Vlasov equation $dX_t = b(t, X_t, [X_t])dt + \\sigma(t, X_t, [X_t])dW_t$ where $[X_t]$ is the law of $X_t$. We specifically consider the kinetic case, where the equation is degenerate because the dimension of the Brownian motion $W$ is strictly smaller than that of the solution $X$, as commonly required in classical models of collisional kinetic theory. Assuming H\\\"older continuous coefficients and a weak H\\\"ormander condition, we prove the well-posedness of the equation. This result advances the existing literature by filling a crucial gap: it addresses the previously un","authors_text":"Alessio Rondelli, Andrea Pascucci","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.PR","submitted_at":"2025-01-19T08:47:32Z","title":"Well-posedness of kinetic McKean-Vlasov equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.10987","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:fade5e11a7b8259bfe4618b1a3f8aca980dff325d398d9f105d656a0e54ad357","target":"record","created_at":"2026-07-05T10:41:19Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"6995c8cbd1f8e848732301104c8ca08ecce39d06a763ae01ecf2b247dee306a2","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.PR","submitted_at":"2025-01-19T08:47:32Z","title_canon_sha256":"f56fa0050bf87988ebf1de2b98514cfa86d33c5d0807f9336c9e2dd44d8ca878"},"schema_version":"1.0","source":{"id":"2501.10987","kind":"arxiv","version":4}},"canonical_sha256":"250d2a10010e3eecd25008698e8508867acbba4991dd995a40017cf6cd564367","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"250d2a10010e3eecd25008698e8508867acbba4991dd995a40017cf6cd564367","first_computed_at":"2026-07-05T10:41:19.747282Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:41:19.747282Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"F3YFv0s0XgseAq9WHhx2TR3OWiHlaX2exsdUecFopSbAsylLy590mYdh92IeykrchwxCW+eLUbwFXEi6wNXsBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T10:41:19.747851Z","signed_message":"canonical_sha256_bytes"},"source_id":"2501.10987","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fade5e11a7b8259bfe4618b1a3f8aca980dff325d398d9f105d656a0e54ad357","sha256:607ee48372b19cae10eea7b835b745f83852b925005fea56abb69d73607eefba"],"state_sha256":"64b439217459a3db65b0e1487882b57ea6ecda54ffcbe5fb227ee998a375728b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U/MLKglkf06dJNr08IY8R2uZi7S9kzyoMn2kP26+EbykMIWhow+qweYeTbXHXwbke17udHeI9bCOPfWSy06FAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-15T12:36:44.144795Z","bundle_sha256":"87e57b7f8d6495f351215f6a14bbd0b80a270aaa62abdf9a206c3c910e4fc985"}}