{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:OBOEMK4PHXDXK55XFKXEG5ZDMB","short_pith_number":"pith:OBOEMK4P","canonical_record":{"source":{"id":"2011.00537","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-11-01T15:30:52Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"c92cc5624dd0eb445a29882aa01f639351de959e56045e52c4d5dfa5001a6acd","abstract_canon_sha256":"f57fd9dfe96703785d0b7592599cd6d5df09ec766df0598cbd2d09d67c08b033"},"schema_version":"1.0"},"canonical_sha256":"705c462b8f3dc77577b72aae43772360545e16ac1ece13925f3c591596217a3b","source":{"kind":"arxiv","id":"2011.00537","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2011.00537","created_at":"2026-07-05T03:42:20Z"},{"alias_kind":"arxiv_version","alias_value":"2011.00537v5","created_at":"2026-07-05T03:42:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2011.00537","created_at":"2026-07-05T03:42:20Z"},{"alias_kind":"pith_short_12","alias_value":"OBOEMK4PHXDX","created_at":"2026-07-05T03:42:20Z"},{"alias_kind":"pith_short_16","alias_value":"OBOEMK4PHXDXK55X","created_at":"2026-07-05T03:42:20Z"},{"alias_kind":"pith_short_8","alias_value":"OBOEMK4P","created_at":"2026-07-05T03:42:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:OBOEMK4PHXDXK55XFKXEG5ZDMB","target":"record","payload":{"canonical_record":{"source":{"id":"2011.00537","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-11-01T15:30:52Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"c92cc5624dd0eb445a29882aa01f639351de959e56045e52c4d5dfa5001a6acd","abstract_canon_sha256":"f57fd9dfe96703785d0b7592599cd6d5df09ec766df0598cbd2d09d67c08b033"},"schema_version":"1.0"},"canonical_sha256":"705c462b8f3dc77577b72aae43772360545e16ac1ece13925f3c591596217a3b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:42:20.093533Z","signature_b64":"VBPpTCVtoYF8ErYh+udlyyBOdPbLYu6AhAGFomKdHvoW0HZkz5pVdJX4dzWnJ7oz3kyIzCdspyAtGMw/dnbLDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"705c462b8f3dc77577b72aae43772360545e16ac1ece13925f3c591596217a3b","last_reissued_at":"2026-07-05T03:42:20.093105Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:42:20.093105Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2011.00537","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T03:42:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wGpN3uogFnld6DNO9tQyn6jyxgR+5OE0QIMbPsRLNWiWYnMosDd+81YD3uN1AtWr7jc/hB23zg4d18q+lw5lAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T06:17:25.635186Z"},"content_sha256":"b3b84f981fd21aa4ad43b6c3c1cfb7a61d9f5acf1af824fcdb6ae3826b70e428","schema_version":"1.0","event_id":"sha256:b3b84f981fd21aa4ad43b6c3c1cfb7a61d9f5acf1af824fcdb6ae3826b70e428"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:OBOEMK4PHXDXK55XFKXEG5ZDMB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Quantitative particle approximation of nonlinear Fokker-Planck equations with singular kernel","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.PR","authors_text":"Alexandre Richard, Christian Olivera, Milica Tomasevic","submitted_at":"2020-11-01T15:30:52Z","abstract_excerpt":"In this work, we study the convergence of the empirical measure of moderately interacting particle systems with singular interaction kernels. First, we prove quantitative convergence of the time marginals of the empirical measure of particle positions towards the solution of the limiting nonlinear Fokker-Planck equation. Second, we prove the well-posedness for the McKean-Vlasov SDE involving such singular kernels and the convergence of the empirical measure towards it (propagation of chaos).\n  Our results only require very weak regularity on the interaction kernel, including the Biot-Savart ke"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2011.00537","kind":"arxiv","version":5},"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/2011.00537/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":"2026-07-05T03:42:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/K8Fq6DH4LgqOTSXcHUVncvWtJkqWcwq2e0D2p8FNaIUznIwAkcHol0r39eGrjntnB4rh/gT3iredyHjzIUoDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T06:17:25.635717Z"},"content_sha256":"f8891ccada7111b55d44cf367f215c8389340b5279da4bfdaea6a2f58728f99d","schema_version":"1.0","event_id":"sha256:f8891ccada7111b55d44cf367f215c8389340b5279da4bfdaea6a2f58728f99d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OBOEMK4PHXDXK55XFKXEG5ZDMB/bundle.json","state_url":"https://pith.science/pith/OBOEMK4PHXDXK55XFKXEG5ZDMB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OBOEMK4PHXDXK55XFKXEG5ZDMB/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":"2026-08-20T06:17:25Z","links":{"resolver":"https://pith.science/pith/OBOEMK4PHXDXK55XFKXEG5ZDMB","bundle":"https://pith.science/pith/OBOEMK4PHXDXK55XFKXEG5ZDMB/bundle.json","state":"https://pith.science/pith/OBOEMK4PHXDXK55XFKXEG5ZDMB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OBOEMK4PHXDXK55XFKXEG5ZDMB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:OBOEMK4PHXDXK55XFKXEG5ZDMB","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":"f57fd9dfe96703785d0b7592599cd6d5df09ec766df0598cbd2d09d67c08b033","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-11-01T15:30:52Z","title_canon_sha256":"c92cc5624dd0eb445a29882aa01f639351de959e56045e52c4d5dfa5001a6acd"},"schema_version":"1.0","source":{"id":"2011.00537","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2011.00537","created_at":"2026-07-05T03:42:20Z"},{"alias_kind":"arxiv_version","alias_value":"2011.00537v5","created_at":"2026-07-05T03:42:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2011.00537","created_at":"2026-07-05T03:42:20Z"},{"alias_kind":"pith_short_12","alias_value":"OBOEMK4PHXDX","created_at":"2026-07-05T03:42:20Z"},{"alias_kind":"pith_short_16","alias_value":"OBOEMK4PHXDXK55X","created_at":"2026-07-05T03:42:20Z"},{"alias_kind":"pith_short_8","alias_value":"OBOEMK4P","created_at":"2026-07-05T03:42:20Z"}],"graph_snapshots":[{"event_id":"sha256:f8891ccada7111b55d44cf367f215c8389340b5279da4bfdaea6a2f58728f99d","target":"graph","created_at":"2026-07-05T03:42:20Z","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/2011.00537/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this work, we study the convergence of the empirical measure of moderately interacting particle systems with singular interaction kernels. First, we prove quantitative convergence of the time marginals of the empirical measure of particle positions towards the solution of the limiting nonlinear Fokker-Planck equation. Second, we prove the well-posedness for the McKean-Vlasov SDE involving such singular kernels and the convergence of the empirical measure towards it (propagation of chaos).\n  Our results only require very weak regularity on the interaction kernel, including the Biot-Savart ke","authors_text":"Alexandre Richard, Christian Olivera, Milica Tomasevic","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-11-01T15:30:52Z","title":"Quantitative particle approximation of nonlinear Fokker-Planck equations with singular kernel"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2011.00537","kind":"arxiv","version":5},"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:b3b84f981fd21aa4ad43b6c3c1cfb7a61d9f5acf1af824fcdb6ae3826b70e428","target":"record","created_at":"2026-07-05T03:42:20Z","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":"f57fd9dfe96703785d0b7592599cd6d5df09ec766df0598cbd2d09d67c08b033","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-11-01T15:30:52Z","title_canon_sha256":"c92cc5624dd0eb445a29882aa01f639351de959e56045e52c4d5dfa5001a6acd"},"schema_version":"1.0","source":{"id":"2011.00537","kind":"arxiv","version":5}},"canonical_sha256":"705c462b8f3dc77577b72aae43772360545e16ac1ece13925f3c591596217a3b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"705c462b8f3dc77577b72aae43772360545e16ac1ece13925f3c591596217a3b","first_computed_at":"2026-07-05T03:42:20.093105Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:42:20.093105Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VBPpTCVtoYF8ErYh+udlyyBOdPbLYu6AhAGFomKdHvoW0HZkz5pVdJX4dzWnJ7oz3kyIzCdspyAtGMw/dnbLDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T03:42:20.093533Z","signed_message":"canonical_sha256_bytes"},"source_id":"2011.00537","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b3b84f981fd21aa4ad43b6c3c1cfb7a61d9f5acf1af824fcdb6ae3826b70e428","sha256:f8891ccada7111b55d44cf367f215c8389340b5279da4bfdaea6a2f58728f99d"],"state_sha256":"448de239b912d4958fb7528af5de1b8d2c9186ea718c7195bdf85d533e0d4eb5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iZ+rrSG7QvevbUl217Nhh4sGGqbsW97g01Sm45/pCp1mtJZLj38MFbAOK45JnXV+EjEwcIpG/cpI4REp4a6SDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-20T06:17:25.640791Z","bundle_sha256":"fc0698eaa1ce86be47999e8bffb676ac73d94862a9b3aa4808811dd5c8959a92"}}