{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:EX4GJ3IKNTUW3PF4DTHC55UZP5","short_pith_number":"pith:EX4GJ3IK","canonical_record":{"source":{"id":"1309.0222","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-09-01T14:06:59Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"ce292dda2f407236fb946e39965cd1ac20fbf70d492a21a38789c736bbcf95a0","abstract_canon_sha256":"588bf49e4c4caa427cca00977fbbd92672780fd45a199e853d2cfa9df1038814"},"schema_version":"1.0"},"canonical_sha256":"25f864ed0a6ce96dbcbc1cce2ef6997f77ee85bb6bfe4d7b6d1131a8f3c52c2d","source":{"kind":"arxiv","id":"1309.0222","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.0222","created_at":"2026-05-18T02:39:24Z"},{"alias_kind":"arxiv_version","alias_value":"1309.0222v2","created_at":"2026-05-18T02:39:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.0222","created_at":"2026-05-18T02:39:24Z"},{"alias_kind":"pith_short_12","alias_value":"EX4GJ3IKNTUW","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"EX4GJ3IKNTUW3PF4","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"EX4GJ3IK","created_at":"2026-05-18T12:27:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:EX4GJ3IKNTUW3PF4DTHC55UZP5","target":"record","payload":{"canonical_record":{"source":{"id":"1309.0222","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-09-01T14:06:59Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"ce292dda2f407236fb946e39965cd1ac20fbf70d492a21a38789c736bbcf95a0","abstract_canon_sha256":"588bf49e4c4caa427cca00977fbbd92672780fd45a199e853d2cfa9df1038814"},"schema_version":"1.0"},"canonical_sha256":"25f864ed0a6ce96dbcbc1cce2ef6997f77ee85bb6bfe4d7b6d1131a8f3c52c2d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:39:24.068568Z","signature_b64":"kbC0HSax81hBL690fyT9mDQ4woj3j92LWYgVE7kymWF/JJ5kGUkp9F5+QUO7mUtXZ5vd/fMHxGVaZsS4s8SzCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"25f864ed0a6ce96dbcbc1cce2ef6997f77ee85bb6bfe4d7b6d1131a8f3c52c2d","last_reissued_at":"2026-05-18T02:39:24.068221Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:39:24.068221Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.0222","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:39:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PwIgvDS3lMaKxF1RP636Wx223XEsx5VVcH/abTYkT9wMmVHoG2ySQSYj8o0+SmbzkXG2RAa6v+MFuWUA3tk6Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T14:40:51.286533Z"},"content_sha256":"e92348668b948e754938b227f6bfd75503b9d4243dd2654be1fa01e0506dc59a","schema_version":"1.0","event_id":"sha256:e92348668b948e754938b227f6bfd75503b9d4243dd2654be1fa01e0506dc59a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:EX4GJ3IKNTUW3PF4DTHC55UZP5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Empirical Measures and Vlasov Hierarchies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Cl\\'ement Mouhot, Fran\\c{c}ois Golse, Valeria Ricci","submitted_at":"2013-09-01T14:06:59Z","abstract_excerpt":"The present note reviews some aspects of the mean field limit for Vlasov type equations with Lipschitz continuous interaction kernel. We discuss in particular the connection between the approach involving the N-particle empirical measure and the formulation based on the BBGKY hierarchy. This leads to a more direct proof of the quantitative estimates on the propagation of chaos obtained on a more general class of interacting systems in [S.Mischler, C. Mouhot, B. Wennberg, arXiv:1101.4727]. Our main result is a stability estimate on the BBGKY hierarchy uniform in the number of particles, which i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.0222","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":""},"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":"2026-05-18T02:39:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PEe8AVR+On6oKTEQg2EhS5me/uBtO5uRBmtmmMWi2iL+wWk6SGDFyqa4ts3XwgM2AcEPnYO2vSYjU4lb+QtJAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T14:40:51.286880Z"},"content_sha256":"f0f3ff21636302164b244ab61e03dae04a269a87f53a92efb5d3eea164c4b92f","schema_version":"1.0","event_id":"sha256:f0f3ff21636302164b244ab61e03dae04a269a87f53a92efb5d3eea164c4b92f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EX4GJ3IKNTUW3PF4DTHC55UZP5/bundle.json","state_url":"https://pith.science/pith/EX4GJ3IKNTUW3PF4DTHC55UZP5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EX4GJ3IKNTUW3PF4DTHC55UZP5/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":"2026-06-01T14:40:51Z","links":{"resolver":"https://pith.science/pith/EX4GJ3IKNTUW3PF4DTHC55UZP5","bundle":"https://pith.science/pith/EX4GJ3IKNTUW3PF4DTHC55UZP5/bundle.json","state":"https://pith.science/pith/EX4GJ3IKNTUW3PF4DTHC55UZP5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EX4GJ3IKNTUW3PF4DTHC55UZP5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:EX4GJ3IKNTUW3PF4DTHC55UZP5","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":"588bf49e4c4caa427cca00977fbbd92672780fd45a199e853d2cfa9df1038814","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-09-01T14:06:59Z","title_canon_sha256":"ce292dda2f407236fb946e39965cd1ac20fbf70d492a21a38789c736bbcf95a0"},"schema_version":"1.0","source":{"id":"1309.0222","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.0222","created_at":"2026-05-18T02:39:24Z"},{"alias_kind":"arxiv_version","alias_value":"1309.0222v2","created_at":"2026-05-18T02:39:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.0222","created_at":"2026-05-18T02:39:24Z"},{"alias_kind":"pith_short_12","alias_value":"EX4GJ3IKNTUW","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"EX4GJ3IKNTUW3PF4","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"EX4GJ3IK","created_at":"2026-05-18T12:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:f0f3ff21636302164b244ab61e03dae04a269a87f53a92efb5d3eea164c4b92f","target":"graph","created_at":"2026-05-18T02:39:24Z","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"},"paper":{"abstract_excerpt":"The present note reviews some aspects of the mean field limit for Vlasov type equations with Lipschitz continuous interaction kernel. We discuss in particular the connection between the approach involving the N-particle empirical measure and the formulation based on the BBGKY hierarchy. This leads to a more direct proof of the quantitative estimates on the propagation of chaos obtained on a more general class of interacting systems in [S.Mischler, C. Mouhot, B. Wennberg, arXiv:1101.4727]. Our main result is a stability estimate on the BBGKY hierarchy uniform in the number of particles, which i","authors_text":"Cl\\'ement Mouhot, Fran\\c{c}ois Golse, Valeria Ricci","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-09-01T14:06:59Z","title":"Empirical Measures and Vlasov Hierarchies"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.0222","kind":"arxiv","version":2},"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:e92348668b948e754938b227f6bfd75503b9d4243dd2654be1fa01e0506dc59a","target":"record","created_at":"2026-05-18T02:39:24Z","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":"588bf49e4c4caa427cca00977fbbd92672780fd45a199e853d2cfa9df1038814","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-09-01T14:06:59Z","title_canon_sha256":"ce292dda2f407236fb946e39965cd1ac20fbf70d492a21a38789c736bbcf95a0"},"schema_version":"1.0","source":{"id":"1309.0222","kind":"arxiv","version":2}},"canonical_sha256":"25f864ed0a6ce96dbcbc1cce2ef6997f77ee85bb6bfe4d7b6d1131a8f3c52c2d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"25f864ed0a6ce96dbcbc1cce2ef6997f77ee85bb6bfe4d7b6d1131a8f3c52c2d","first_computed_at":"2026-05-18T02:39:24.068221Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:39:24.068221Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kbC0HSax81hBL690fyT9mDQ4woj3j92LWYgVE7kymWF/JJ5kGUkp9F5+QUO7mUtXZ5vd/fMHxGVaZsS4s8SzCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:39:24.068568Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.0222","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e92348668b948e754938b227f6bfd75503b9d4243dd2654be1fa01e0506dc59a","sha256:f0f3ff21636302164b244ab61e03dae04a269a87f53a92efb5d3eea164c4b92f"],"state_sha256":"7b13b38703ca804f5ddf89ca520131ca8d3576851dbe2d2c5230e3f29c6327e0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kBGcbkWC6Fl+zYFr3kEGk7w6bJNM7I8cGl3ZhAJocB/jyRozmndVHsJqcseXERXSGV6MAS6GzprbtI6E6bEXAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T14:40:51.288798Z","bundle_sha256":"9e9bc9582d994a27d5c990b323c74ddb7f108b93022d98dde3318b3373f8f9b2"}}