{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:4RUKVIO2Y6WPG7ALRHFRCWDHQB","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":"52018b0b4d8e58efd41f33763366e057c7813d7cc32e537a99a3b2d61993330b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-12-16T10:54:43Z","title_canon_sha256":"3b4257e096871014226bb54a7bfa674e3277c0c2ff6eae5f5ee871e0931b5ac9"},"schema_version":"1.0","source":{"id":"1612.05425","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.05425","created_at":"2026-05-18T00:38:41Z"},{"alias_kind":"arxiv_version","alias_value":"1612.05425v3","created_at":"2026-05-18T00:38:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.05425","created_at":"2026-05-18T00:38:41Z"},{"alias_kind":"pith_short_12","alias_value":"4RUKVIO2Y6WP","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"4RUKVIO2Y6WPG7AL","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"4RUKVIO2","created_at":"2026-05-18T12:29:58Z"}],"graph_snapshots":[{"event_id":"sha256:1cc396eb35a56572ec6af5316bee8cc615792aab659dc38b7bee6834551fb8b2","target":"graph","created_at":"2026-05-18T00:38:41Z","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":"This paper establishes the existence of relaxed solutions to mean field games (MFGs for short) with singular controls. We also prove approximations of solutions results for a particular class of MFGs with singular controls by solutions, respectively control rules, for MFGs with purely regular controls. Our existence and approximation results strongly hinge on the use of the Skorokhod $M_1$ topology on the space of c\\`adl\\`ag functions.","authors_text":"Guanxing Fu, Ulrich Horst","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-12-16T10:54:43Z","title":"Mean Field Games with Singular Controls"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.05425","kind":"arxiv","version":3},"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":"sha256:657ced0c2d875c9f286bbbc8cdae2673b6ffd0937b0b06b01b8d0f6e566ead07","target":"record","created_at":"2026-05-18T00:38:41Z","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":"52018b0b4d8e58efd41f33763366e057c7813d7cc32e537a99a3b2d61993330b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-12-16T10:54:43Z","title_canon_sha256":"3b4257e096871014226bb54a7bfa674e3277c0c2ff6eae5f5ee871e0931b5ac9"},"schema_version":"1.0","source":{"id":"1612.05425","kind":"arxiv","version":3}},"canonical_sha256":"e468aaa1dac7acf37c0b89cb1158678074be94a62acddac73b5f4953f5f7cc97","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e468aaa1dac7acf37c0b89cb1158678074be94a62acddac73b5f4953f5f7cc97","first_computed_at":"2026-05-18T00:38:41.892855Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:38:41.892855Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yIK2qyVyhXyvQQ7O1e/YP0lfZvHqAXB+UBVcYFcIPe4X4pzJJkvQVb1zZs3uX80ULSUkeuvLzTlWyOelEjBaAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:38:41.893531Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.05425","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:657ced0c2d875c9f286bbbc8cdae2673b6ffd0937b0b06b01b8d0f6e566ead07","sha256:1cc396eb35a56572ec6af5316bee8cc615792aab659dc38b7bee6834551fb8b2"],"state_sha256":"5dc97a31f13b4606f5d2411799efdbf1ad78a933b05323769ff0f3e593ce3449"}