{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:X74TZUYH3PEEEBDOVPDT65XBCF","short_pith_number":"pith:X74TZUYH","canonical_record":{"source":{"id":"2506.11900","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-06-13T15:51:53Z","cross_cats_sorted":[],"title_canon_sha256":"bace71950c448be2736ae6f56c7997173f231fb50513bfe93506e753387162ed","abstract_canon_sha256":"31aa2edc744f063cafb4b4b5a773c6d26a9ef6037d90a9f6cd553fd68cc0b918"},"schema_version":"1.0"},"canonical_sha256":"bff93cd307dbc842046eabc73f76e111657d9a107c879efccc19f2e955d603c3","source":{"kind":"arxiv","id":"2506.11900","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.11900","created_at":"2026-07-05T11:21:12Z"},{"alias_kind":"arxiv_version","alias_value":"2506.11900v1","created_at":"2026-07-05T11:21:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.11900","created_at":"2026-07-05T11:21:12Z"},{"alias_kind":"pith_short_12","alias_value":"X74TZUYH3PEE","created_at":"2026-07-05T11:21:12Z"},{"alias_kind":"pith_short_16","alias_value":"X74TZUYH3PEEEBDO","created_at":"2026-07-05T11:21:12Z"},{"alias_kind":"pith_short_8","alias_value":"X74TZUYH","created_at":"2026-07-05T11:21:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:X74TZUYH3PEEEBDOVPDT65XBCF","target":"record","payload":{"canonical_record":{"source":{"id":"2506.11900","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-06-13T15:51:53Z","cross_cats_sorted":[],"title_canon_sha256":"bace71950c448be2736ae6f56c7997173f231fb50513bfe93506e753387162ed","abstract_canon_sha256":"31aa2edc744f063cafb4b4b5a773c6d26a9ef6037d90a9f6cd553fd68cc0b918"},"schema_version":"1.0"},"canonical_sha256":"bff93cd307dbc842046eabc73f76e111657d9a107c879efccc19f2e955d603c3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:21:12.519403Z","signature_b64":"eXjBcbjD5u/F+iDvQzSiskAxVfV4ibfTjmv46362O7SmY2ifDTDA7oofsFEnoSvyI+2qRpRvu/6ZenQdk0LCBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bff93cd307dbc842046eabc73f76e111657d9a107c879efccc19f2e955d603c3","last_reissued_at":"2026-07-05T11:21:12.518862Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:21:12.518862Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2506.11900","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:21:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EOjpJZkTu0/NVpmN2SbMbgS1gRc2EOkifX/o3BXwv9NMquvJ+dlL6UjZV3p3lOfWP9QXnKUNjyQSQPBMdJ3XDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T17:52:07.002704Z"},"content_sha256":"2ae6ca349322f6ed37373e4f99f1627059e0ba46969715eb3d3da0b8c6b5cde9","schema_version":"1.0","event_id":"sha256:2ae6ca349322f6ed37373e4f99f1627059e0ba46969715eb3d3da0b8c6b5cde9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:X74TZUYH3PEEEBDOVPDT65XBCF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the density of singular SDEs with fractional noise and applications to McKean-Vlasov equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alexandre Richard, Etienne Tanr\\'e, Lucio Galeati, Lukas Anzeletti","submitted_at":"2025-06-13T15:51:53Z","abstract_excerpt":"We investigate properties of the (conditional) law of the solution to SDEs driven by fractional Brownian noise with a singular, possibly distributional, drift. Our results on the law are twofold: i) we quantify the spatial regularity of the law, while keeping track of integrability in time, and ii) we prove that it has a density with Gaussian tails. Then the former result is used to establish novel results on existence and uniqueness of solutions to McKean-Vlasov equations of convolutional type."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.11900","kind":"arxiv","version":1},"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/2506.11900/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":"2026-07-05T11:21:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XXVxd3JqIUshj6CbN/+m3ZSAwIunbCkA4Lb6W3wBY03EHBHS6duTOIqww920Gfxg0JLT2A5Jwh0PF9Z8KiOMBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T17:52:07.003256Z"},"content_sha256":"717e373339424d8ffd43302c2500683f1dc710161f754aaca3e464bc6d2a4039","schema_version":"1.0","event_id":"sha256:717e373339424d8ffd43302c2500683f1dc710161f754aaca3e464bc6d2a4039"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/X74TZUYH3PEEEBDOVPDT65XBCF/bundle.json","state_url":"https://pith.science/pith/X74TZUYH3PEEEBDOVPDT65XBCF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/X74TZUYH3PEEEBDOVPDT65XBCF/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":"2026-08-08T17:52:07Z","links":{"resolver":"https://pith.science/pith/X74TZUYH3PEEEBDOVPDT65XBCF","bundle":"https://pith.science/pith/X74TZUYH3PEEEBDOVPDT65XBCF/bundle.json","state":"https://pith.science/pith/X74TZUYH3PEEEBDOVPDT65XBCF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/X74TZUYH3PEEEBDOVPDT65XBCF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:X74TZUYH3PEEEBDOVPDT65XBCF","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":"31aa2edc744f063cafb4b4b5a773c6d26a9ef6037d90a9f6cd553fd68cc0b918","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-06-13T15:51:53Z","title_canon_sha256":"bace71950c448be2736ae6f56c7997173f231fb50513bfe93506e753387162ed"},"schema_version":"1.0","source":{"id":"2506.11900","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.11900","created_at":"2026-07-05T11:21:12Z"},{"alias_kind":"arxiv_version","alias_value":"2506.11900v1","created_at":"2026-07-05T11:21:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.11900","created_at":"2026-07-05T11:21:12Z"},{"alias_kind":"pith_short_12","alias_value":"X74TZUYH3PEE","created_at":"2026-07-05T11:21:12Z"},{"alias_kind":"pith_short_16","alias_value":"X74TZUYH3PEEEBDO","created_at":"2026-07-05T11:21:12Z"},{"alias_kind":"pith_short_8","alias_value":"X74TZUYH","created_at":"2026-07-05T11:21:12Z"}],"graph_snapshots":[{"event_id":"sha256:717e373339424d8ffd43302c2500683f1dc710161f754aaca3e464bc6d2a4039","target":"graph","created_at":"2026-07-05T11:21:12Z","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/2506.11900/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We investigate properties of the (conditional) law of the solution to SDEs driven by fractional Brownian noise with a singular, possibly distributional, drift. Our results on the law are twofold: i) we quantify the spatial regularity of the law, while keeping track of integrability in time, and ii) we prove that it has a density with Gaussian tails. Then the former result is used to establish novel results on existence and uniqueness of solutions to McKean-Vlasov equations of convolutional type.","authors_text":"Alexandre Richard, Etienne Tanr\\'e, Lucio Galeati, Lukas Anzeletti","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-06-13T15:51:53Z","title":"On the density of singular SDEs with fractional noise and applications to McKean-Vlasov equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.11900","kind":"arxiv","version":1},"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":"sha256:2ae6ca349322f6ed37373e4f99f1627059e0ba46969715eb3d3da0b8c6b5cde9","target":"record","created_at":"2026-07-05T11:21:12Z","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":"31aa2edc744f063cafb4b4b5a773c6d26a9ef6037d90a9f6cd553fd68cc0b918","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-06-13T15:51:53Z","title_canon_sha256":"bace71950c448be2736ae6f56c7997173f231fb50513bfe93506e753387162ed"},"schema_version":"1.0","source":{"id":"2506.11900","kind":"arxiv","version":1}},"canonical_sha256":"bff93cd307dbc842046eabc73f76e111657d9a107c879efccc19f2e955d603c3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bff93cd307dbc842046eabc73f76e111657d9a107c879efccc19f2e955d603c3","first_computed_at":"2026-07-05T11:21:12.518862Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:21:12.518862Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eXjBcbjD5u/F+iDvQzSiskAxVfV4ibfTjmv46362O7SmY2ifDTDA7oofsFEnoSvyI+2qRpRvu/6ZenQdk0LCBA==","signature_status":"signed_v1","signed_at":"2026-07-05T11:21:12.519403Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.11900","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2ae6ca349322f6ed37373e4f99f1627059e0ba46969715eb3d3da0b8c6b5cde9","sha256:717e373339424d8ffd43302c2500683f1dc710161f754aaca3e464bc6d2a4039"],"state_sha256":"c9a15f0cf06ce78051408e4955de3fd3507af85fdefd59fd2778bd3b24508692"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LMps9IDfx5lyEIsctQfFfSlDUUvqSxjon3DDFyRZ3BZaQl6O/6HKFxE9g2Z6SdF9b9FuR06KR8K6C3yGmIXGAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T17:52:07.008129Z","bundle_sha256":"993b0566f3b71ae488c80dfd323d398e465f167e5dbcf0426722c3fa6a5d5c28"}}