{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:7RE5LLX42MKHPOWQLSHVIUEI2P","short_pith_number":"pith:7RE5LLX4","canonical_record":{"source":{"id":"1103.5186","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-03-27T06:12:17Z","cross_cats_sorted":[],"title_canon_sha256":"ac1e74103b95375feb83431002e30875c6accba58f487b533b601c34533096bc","abstract_canon_sha256":"1214bb1f92d3bcdf6b1503856f9ac5289c16524fa8658de51ff28abae074f911"},"schema_version":"1.0"},"canonical_sha256":"fc49d5aefcd31477bad05c8f545088d3edc96a2d30de7bbdd7d5d9676f1ed840","source":{"kind":"arxiv","id":"1103.5186","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.5186","created_at":"2026-05-18T04:25:38Z"},{"alias_kind":"arxiv_version","alias_value":"1103.5186v1","created_at":"2026-05-18T04:25:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.5186","created_at":"2026-05-18T04:25:38Z"},{"alias_kind":"pith_short_12","alias_value":"7RE5LLX42MKH","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"7RE5LLX42MKHPOWQ","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"7RE5LLX4","created_at":"2026-05-18T12:26:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:7RE5LLX42MKHPOWQLSHVIUEI2P","target":"record","payload":{"canonical_record":{"source":{"id":"1103.5186","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-03-27T06:12:17Z","cross_cats_sorted":[],"title_canon_sha256":"ac1e74103b95375feb83431002e30875c6accba58f487b533b601c34533096bc","abstract_canon_sha256":"1214bb1f92d3bcdf6b1503856f9ac5289c16524fa8658de51ff28abae074f911"},"schema_version":"1.0"},"canonical_sha256":"fc49d5aefcd31477bad05c8f545088d3edc96a2d30de7bbdd7d5d9676f1ed840","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:25:38.463433Z","signature_b64":"aGU7But4F8sxWS1Pm70h/3hW51ggPZbC5w1johcGoTFgPq3KCsNTGEhwoMxqTfpFtVoWWQEc7d7UNpiO+8DwCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fc49d5aefcd31477bad05c8f545088d3edc96a2d30de7bbdd7d5d9676f1ed840","last_reissued_at":"2026-05-18T04:25:38.462980Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:25:38.462980Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1103.5186","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":"2026-05-18T04:25:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WCYK95DSh3jAqbXWTZ4bdqM/sWX/wufMvEY8EFZNgui8LzviTJSzPIcQptAxwWK6JgQ8IL5GWVaQRs/EacRRDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T22:20:54.289882Z"},"content_sha256":"38989a41f7c79e309d9695715d2067dd907579d5a875726f4468711f3ff1d819","schema_version":"1.0","event_id":"sha256:38989a41f7c79e309d9695715d2067dd907579d5a875726f4468711f3ff1d819"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:7RE5LLX42MKHPOWQLSHVIUEI2P","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Invariance measures of stochastic 2D Navier-Stokes equations driven by $\\alpha$-stable processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Lihu Xu, Xicheng Zhang, Zhao Dong","submitted_at":"2011-03-27T06:12:17Z","abstract_excerpt":"In this note we prove the well-posedness for stochastic 2D Navier-Stokes equation driven by general L\\'evy processes (in particular, $\\alpha$-stable processes), and obtain the existence of invariant measures."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.5186","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":""},"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":"2026-05-18T04:25:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"98A6Ju4h7LmAdjrxar2m4IWxU6CLvt7nzrTQxJO8cmkleG/adTZw88T7vZy6jZNWZwTXetpcQnEaqHPehgsdAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T22:20:54.290467Z"},"content_sha256":"03034bedaed49677dae88be869bd7e9f67cc1609924ecb24fc0c32c2932a8d77","schema_version":"1.0","event_id":"sha256:03034bedaed49677dae88be869bd7e9f67cc1609924ecb24fc0c32c2932a8d77"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7RE5LLX42MKHPOWQLSHVIUEI2P/bundle.json","state_url":"https://pith.science/pith/7RE5LLX42MKHPOWQLSHVIUEI2P/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7RE5LLX42MKHPOWQLSHVIUEI2P/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":"2026-06-04T22:20:54Z","links":{"resolver":"https://pith.science/pith/7RE5LLX42MKHPOWQLSHVIUEI2P","bundle":"https://pith.science/pith/7RE5LLX42MKHPOWQLSHVIUEI2P/bundle.json","state":"https://pith.science/pith/7RE5LLX42MKHPOWQLSHVIUEI2P/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7RE5LLX42MKHPOWQLSHVIUEI2P/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:7RE5LLX42MKHPOWQLSHVIUEI2P","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":"1214bb1f92d3bcdf6b1503856f9ac5289c16524fa8658de51ff28abae074f911","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-03-27T06:12:17Z","title_canon_sha256":"ac1e74103b95375feb83431002e30875c6accba58f487b533b601c34533096bc"},"schema_version":"1.0","source":{"id":"1103.5186","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.5186","created_at":"2026-05-18T04:25:38Z"},{"alias_kind":"arxiv_version","alias_value":"1103.5186v1","created_at":"2026-05-18T04:25:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.5186","created_at":"2026-05-18T04:25:38Z"},{"alias_kind":"pith_short_12","alias_value":"7RE5LLX42MKH","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"7RE5LLX42MKHPOWQ","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"7RE5LLX4","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:03034bedaed49677dae88be869bd7e9f67cc1609924ecb24fc0c32c2932a8d77","target":"graph","created_at":"2026-05-18T04:25:38Z","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 this note we prove the well-posedness for stochastic 2D Navier-Stokes equation driven by general L\\'evy processes (in particular, $\\alpha$-stable processes), and obtain the existence of invariant measures.","authors_text":"Lihu Xu, Xicheng Zhang, Zhao Dong","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-03-27T06:12:17Z","title":"Invariance measures of stochastic 2D Navier-Stokes equations driven by $\\alpha$-stable processes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.5186","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":"sha256:38989a41f7c79e309d9695715d2067dd907579d5a875726f4468711f3ff1d819","target":"record","created_at":"2026-05-18T04:25:38Z","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":"1214bb1f92d3bcdf6b1503856f9ac5289c16524fa8658de51ff28abae074f911","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-03-27T06:12:17Z","title_canon_sha256":"ac1e74103b95375feb83431002e30875c6accba58f487b533b601c34533096bc"},"schema_version":"1.0","source":{"id":"1103.5186","kind":"arxiv","version":1}},"canonical_sha256":"fc49d5aefcd31477bad05c8f545088d3edc96a2d30de7bbdd7d5d9676f1ed840","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fc49d5aefcd31477bad05c8f545088d3edc96a2d30de7bbdd7d5d9676f1ed840","first_computed_at":"2026-05-18T04:25:38.462980Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:25:38.462980Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aGU7But4F8sxWS1Pm70h/3hW51ggPZbC5w1johcGoTFgPq3KCsNTGEhwoMxqTfpFtVoWWQEc7d7UNpiO+8DwCA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:25:38.463433Z","signed_message":"canonical_sha256_bytes"},"source_id":"1103.5186","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:38989a41f7c79e309d9695715d2067dd907579d5a875726f4468711f3ff1d819","sha256:03034bedaed49677dae88be869bd7e9f67cc1609924ecb24fc0c32c2932a8d77"],"state_sha256":"a9f720410be428f9e5f04c46e1d2bcdbdd7a762164742d6a4151a1471198aaa5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rd1rYH7XptY3M32fYaSwAmjTGxv6cdgBTVZN8AaCD4qnFKqdYkCkYodZyzkatGbvRUkOhcY+XdOleo8SMtksCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T22:20:54.293706Z","bundle_sha256":"7c06662d63cfeee5976051544a0fde6016ada54992443bbfbcc19985f1c87d07"}}