{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:XMW4HQJMHS2X5YOBJ6GFYIT7ED","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":"032df9ab71d14d6061a47c2feb5872edcc2ec9a05b9cbe2386b407c1f9f4c967","cross_cats_sorted":["math.DS","math.OC","math.PR","q-fin.MF"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-04-16T06:50:27Z","title_canon_sha256":"0999bf3865ad7b2a7adf0ba17bb199a5db6f2efc5bc2be7f17e454da09c13899"},"schema_version":"1.0","source":{"id":"1404.4150","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1404.4150","created_at":"2026-05-18T02:38:37Z"},{"alias_kind":"arxiv_version","alias_value":"1404.4150v2","created_at":"2026-05-18T02:38:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1404.4150","created_at":"2026-05-18T02:38:37Z"},{"alias_kind":"pith_short_12","alias_value":"XMW4HQJMHS2X","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_16","alias_value":"XMW4HQJMHS2X5YOB","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_8","alias_value":"XMW4HQJM","created_at":"2026-05-18T12:28:57Z"}],"graph_snapshots":[{"event_id":"sha256:faa2acf804550e07433b29a53fb1ca1277a38c01a1a428378df7e0df77bb3db3","target":"graph","created_at":"2026-05-18T02:38:37Z","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":"In his lectures at College de France, P.L. Lions introduced the concept of Master equation, see [5] for Mean Field Games. It is introduced in a heuristic fashion, from the system of partial differential equations, associated to a Nash equilibrium for a large, but finite, number of players. The method, also explained in[2], consists in a formal analogy of terms. The interest of this equation is that it contains interesting particular cases, which can be studied directly, in particular the system of HJB-FP (Hamilton-Jacobi-Bellman, Fokker-Planck) equations obtained as the limit of the finite Nas","authors_text":"Alain Bensoussan, Jens Frehse, Phillip Yam","cross_cats":["math.DS","math.OC","math.PR","q-fin.MF"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-04-16T06:50:27Z","title":"The Master Equation in Mean Field Theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.4150","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:e671a4f316d259f1662845e66a2d036f8218393102d2cbefd122648f97bf71c9","target":"record","created_at":"2026-05-18T02:38:37Z","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":"032df9ab71d14d6061a47c2feb5872edcc2ec9a05b9cbe2386b407c1f9f4c967","cross_cats_sorted":["math.DS","math.OC","math.PR","q-fin.MF"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-04-16T06:50:27Z","title_canon_sha256":"0999bf3865ad7b2a7adf0ba17bb199a5db6f2efc5bc2be7f17e454da09c13899"},"schema_version":"1.0","source":{"id":"1404.4150","kind":"arxiv","version":2}},"canonical_sha256":"bb2dc3c12c3cb57ee1c14f8c5c227f20ec5728da785e43bfc5bba178eeec2c07","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bb2dc3c12c3cb57ee1c14f8c5c227f20ec5728da785e43bfc5bba178eeec2c07","first_computed_at":"2026-05-18T02:38:37.061421Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:38:37.061421Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Elfw1+1ABgZeR8fE/7Ot7uI/c0sfTrUgj6A+6obWGICR7JxjYBliZAlnNii7H3+QB0/c3i6zRLvLgnBFLiLTDg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:38:37.062078Z","signed_message":"canonical_sha256_bytes"},"source_id":"1404.4150","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e671a4f316d259f1662845e66a2d036f8218393102d2cbefd122648f97bf71c9","sha256:faa2acf804550e07433b29a53fb1ca1277a38c01a1a428378df7e0df77bb3db3"],"state_sha256":"12a5749959d006adf0cd2c9bf695659789eb04f38ca2f58171f047c62cc55499"}