{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2005:5Q76DSTEWHVSWABSGMPV7H3EZ4","short_pith_number":"pith:5Q76DSTE","canonical_record":{"source":{"id":"math/0512361","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2005-12-15T13:57:10Z","cross_cats_sorted":["cs.NA","math.NA"],"title_canon_sha256":"0cc7c54e9273bd94aecaf84725630507779d18a45f01eb2c9b1db8c828f2a110","abstract_canon_sha256":"5f229b0d64f3ab42bbad93345dc2852dd8e75b6577512e0a9badec926f4ceef2"},"schema_version":"1.0"},"canonical_sha256":"ec3fe1ca64b1eb2b0032331f5f9f64cf3c97fee4616b3140f64320d5839e587c","source":{"kind":"arxiv","id":"math/0512361","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0512361","created_at":"2026-06-03T22:06:18Z"},{"alias_kind":"arxiv_version","alias_value":"math/0512361v2","created_at":"2026-06-03T22:06:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0512361","created_at":"2026-06-03T22:06:18Z"},{"alias_kind":"pith_short_12","alias_value":"5Q76DSTEWHVS","created_at":"2026-06-03T22:06:18Z"},{"alias_kind":"pith_short_16","alias_value":"5Q76DSTEWHVSWABS","created_at":"2026-06-03T22:06:18Z"},{"alias_kind":"pith_short_8","alias_value":"5Q76DSTE","created_at":"2026-06-03T22:06:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2005:5Q76DSTEWHVSWABSGMPV7H3EZ4","target":"record","payload":{"canonical_record":{"source":{"id":"math/0512361","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2005-12-15T13:57:10Z","cross_cats_sorted":["cs.NA","math.NA"],"title_canon_sha256":"0cc7c54e9273bd94aecaf84725630507779d18a45f01eb2c9b1db8c828f2a110","abstract_canon_sha256":"5f229b0d64f3ab42bbad93345dc2852dd8e75b6577512e0a9badec926f4ceef2"},"schema_version":"1.0"},"canonical_sha256":"ec3fe1ca64b1eb2b0032331f5f9f64cf3c97fee4616b3140f64320d5839e587c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T22:06:18.512910Z","signature_b64":"KV+0XvNEB4M7xm8XbJgE6hUar+0yacROlME6K6lKokCoe2MGPo812v+7puoYkXYExyWGQxXVBkIAB6FYf7rwDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ec3fe1ca64b1eb2b0032331f5f9f64cf3c97fee4616b3140f64320d5839e587c","last_reissued_at":"2026-06-03T22:06:18.512546Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T22:06:18.512546Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0512361","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":"2026-06-03T22:06:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7k13+zgWHHl30EAasKkrZ6rbDsTQLf03DYrN1S41r4JBW0SaT1pPBX4UC5uzqbog6EKQMBRf4YnOGwzJkBd3Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T04:06:17.718406Z"},"content_sha256":"d06fa84a99f1311c180228fd7786acc38d8b2ca12b016abe850f88da96f7e97b","schema_version":"1.0","event_id":"sha256:d06fa84a99f1311c180228fd7786acc38d8b2ca12b016abe850f88da96f7e97b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2005:5Q76DSTEWHVSWABSGMPV7H3EZ4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Markov solutions for the 3D stochastic Navier--Stokes equations with state dependent noise","license":"","headline":"","cross_cats":["cs.NA","math.NA"],"primary_cat":"math.AP","authors_text":"Arnaud Debussche (IRMAR), Cyril Odasso (IRMAR)","submitted_at":"2005-12-15T13:57:10Z","abstract_excerpt":"We construct a Markov family of solutions for the 3D Navier-Stokes equation perturbed by a non degenerate noise. We improve the result of [DPD-NS3D] in two directions. We see that in fact not only a transition semigroup but a Markov family of solutions can be constructed. Moreover, we consider a state dependant noise. Another feature of this work is that we greatly simplify the proofs of [DPD-NS3D]."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0512361","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0512361/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":"2026-06-03T22:06:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CZx7qC+lkz9nya/a6VQmWXWF+ems8GSZco8Ed8BheRm+ZKK8ePJs0JuN37DB2MRBuAhHWyN72drrnv0Fg4LFDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T04:06:17.718965Z"},"content_sha256":"af927c1b596aa49615a709d2bce3956b19760fc4a0ed30f15045496b0175b6c0","schema_version":"1.0","event_id":"sha256:af927c1b596aa49615a709d2bce3956b19760fc4a0ed30f15045496b0175b6c0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5Q76DSTEWHVSWABSGMPV7H3EZ4/bundle.json","state_url":"https://pith.science/pith/5Q76DSTEWHVSWABSGMPV7H3EZ4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5Q76DSTEWHVSWABSGMPV7H3EZ4/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":"2026-06-05T04:06:17Z","links":{"resolver":"https://pith.science/pith/5Q76DSTEWHVSWABSGMPV7H3EZ4","bundle":"https://pith.science/pith/5Q76DSTEWHVSWABSGMPV7H3EZ4/bundle.json","state":"https://pith.science/pith/5Q76DSTEWHVSWABSGMPV7H3EZ4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5Q76DSTEWHVSWABSGMPV7H3EZ4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2005:5Q76DSTEWHVSWABSGMPV7H3EZ4","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":"5f229b0d64f3ab42bbad93345dc2852dd8e75b6577512e0a9badec926f4ceef2","cross_cats_sorted":["cs.NA","math.NA"],"license":"","primary_cat":"math.AP","submitted_at":"2005-12-15T13:57:10Z","title_canon_sha256":"0cc7c54e9273bd94aecaf84725630507779d18a45f01eb2c9b1db8c828f2a110"},"schema_version":"1.0","source":{"id":"math/0512361","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0512361","created_at":"2026-06-03T22:06:18Z"},{"alias_kind":"arxiv_version","alias_value":"math/0512361v2","created_at":"2026-06-03T22:06:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0512361","created_at":"2026-06-03T22:06:18Z"},{"alias_kind":"pith_short_12","alias_value":"5Q76DSTEWHVS","created_at":"2026-06-03T22:06:18Z"},{"alias_kind":"pith_short_16","alias_value":"5Q76DSTEWHVSWABS","created_at":"2026-06-03T22:06:18Z"},{"alias_kind":"pith_short_8","alias_value":"5Q76DSTE","created_at":"2026-06-03T22:06:18Z"}],"graph_snapshots":[{"event_id":"sha256:af927c1b596aa49615a709d2bce3956b19760fc4a0ed30f15045496b0175b6c0","target":"graph","created_at":"2026-06-03T22:06:18Z","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/math/0512361/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We construct a Markov family of solutions for the 3D Navier-Stokes equation perturbed by a non degenerate noise. We improve the result of [DPD-NS3D] in two directions. We see that in fact not only a transition semigroup but a Markov family of solutions can be constructed. Moreover, we consider a state dependant noise. Another feature of this work is that we greatly simplify the proofs of [DPD-NS3D].","authors_text":"Arnaud Debussche (IRMAR), Cyril Odasso (IRMAR)","cross_cats":["cs.NA","math.NA"],"headline":"","license":"","primary_cat":"math.AP","submitted_at":"2005-12-15T13:57:10Z","title":"Markov solutions for the 3D stochastic Navier--Stokes equations with state dependent noise"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0512361","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:d06fa84a99f1311c180228fd7786acc38d8b2ca12b016abe850f88da96f7e97b","target":"record","created_at":"2026-06-03T22:06:18Z","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":"5f229b0d64f3ab42bbad93345dc2852dd8e75b6577512e0a9badec926f4ceef2","cross_cats_sorted":["cs.NA","math.NA"],"license":"","primary_cat":"math.AP","submitted_at":"2005-12-15T13:57:10Z","title_canon_sha256":"0cc7c54e9273bd94aecaf84725630507779d18a45f01eb2c9b1db8c828f2a110"},"schema_version":"1.0","source":{"id":"math/0512361","kind":"arxiv","version":2}},"canonical_sha256":"ec3fe1ca64b1eb2b0032331f5f9f64cf3c97fee4616b3140f64320d5839e587c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ec3fe1ca64b1eb2b0032331f5f9f64cf3c97fee4616b3140f64320d5839e587c","first_computed_at":"2026-06-03T22:06:18.512546Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-03T22:06:18.512546Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KV+0XvNEB4M7xm8XbJgE6hUar+0yacROlME6K6lKokCoe2MGPo812v+7puoYkXYExyWGQxXVBkIAB6FYf7rwDA==","signature_status":"signed_v1","signed_at":"2026-06-03T22:06:18.512910Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0512361","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d06fa84a99f1311c180228fd7786acc38d8b2ca12b016abe850f88da96f7e97b","sha256:af927c1b596aa49615a709d2bce3956b19760fc4a0ed30f15045496b0175b6c0"],"state_sha256":"66b9b246cda46c3974504770030bbbde38e63e9c09866a43d3a10342862cae21"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"woScFHws41AJH8/t6KFrL295Umj+ET5sdOW7QKxfSyw9A8h1NTAx/z/ZIpCMhEhpvE7AarvWbjwSHcNM4ijPBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T04:06:17.722500Z","bundle_sha256":"837319716aaba49a0d2d16f5273a369ab6b81db149cb204ed081aba7f08912b7"}}