{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:G7FGUOY4UMKLWTGUMYMMNNGWQM","short_pith_number":"pith:G7FGUOY4","canonical_record":{"source":{"id":"1409.3430","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-09-11T13:31:31Z","cross_cats_sorted":[],"title_canon_sha256":"3f43f39b88c06c94e89f3096e81b8fc013377d5414595b712fcfc564ae186ffc","abstract_canon_sha256":"8719a307aaaf9127249a9edb1e8092995349d995dcfc74775678723aacb0c8d1"},"schema_version":"1.0"},"canonical_sha256":"37ca6a3b1ca314bb4cd46618c6b4d6830c1f7d550c7699d694043a1eab6a1945","source":{"kind":"arxiv","id":"1409.3430","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.3430","created_at":"2026-05-18T02:43:01Z"},{"alias_kind":"arxiv_version","alias_value":"1409.3430v1","created_at":"2026-05-18T02:43:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.3430","created_at":"2026-05-18T02:43:01Z"},{"alias_kind":"pith_short_12","alias_value":"G7FGUOY4UMKL","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"G7FGUOY4UMKLWTGU","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"G7FGUOY4","created_at":"2026-05-18T12:28:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:G7FGUOY4UMKLWTGUMYMMNNGWQM","target":"record","payload":{"canonical_record":{"source":{"id":"1409.3430","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-09-11T13:31:31Z","cross_cats_sorted":[],"title_canon_sha256":"3f43f39b88c06c94e89f3096e81b8fc013377d5414595b712fcfc564ae186ffc","abstract_canon_sha256":"8719a307aaaf9127249a9edb1e8092995349d995dcfc74775678723aacb0c8d1"},"schema_version":"1.0"},"canonical_sha256":"37ca6a3b1ca314bb4cd46618c6b4d6830c1f7d550c7699d694043a1eab6a1945","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:43:01.396910Z","signature_b64":"bKOUC2CGoesQMClF8qmxw07AWI4P+Hnw9yzEQFNNSrLNcgQ6fF1Uxc2wgQazZVfpynLJW/e1mPUUJCvGoS5HDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"37ca6a3b1ca314bb4cd46618c6b4d6830c1f7d550c7699d694043a1eab6a1945","last_reissued_at":"2026-05-18T02:43:01.396282Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:43:01.396282Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1409.3430","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":"2026-05-18T02:43:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"z2G9JQMLP8R/c/ExiF4/64/ogVUxdQ+zsSSMd8xyTt5thFKmm9kA33EJxzRSSGQBW2g7Dd1otQdxXpEUPipiDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T21:35:37.681569Z"},"content_sha256":"261da63868100b39a0e4ea8b691b11b8297dd4d1041dd7ea36d94f0133f5194b","schema_version":"1.0","event_id":"sha256:261da63868100b39a0e4ea8b691b11b8297dd4d1041dd7ea36d94f0133f5194b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:G7FGUOY4UMKLWTGUMYMMNNGWQM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Invariant and ergodic measures for G-diffusion processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Falei Wang, Guoqiang Zheng, Hanwu Li, Mingshang Hu","submitted_at":"2014-09-11T13:31:31Z","abstract_excerpt":"In this paper we study the problems of invariant and ergodic measures under G-expectation framework. In particular, the stochastic differential equations driven by G-Brownian motion have the unique invariant and ergodic measures. Moreover, the invariant and ergodic measures of G-SDEs are also sublinear expectations. However, the invariant measures may not coincide with ergodic measures, which is different from the classical case."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.3430","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":"2026-05-18T02:43:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ak+CZPF70C+fKjuhzXd9lkckWWroqOrsRZD/KDHKc8ARIxrW+6kU08sEmPro9Ihb4GstctcUWfRd6n7KHncCAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T21:35:37.682193Z"},"content_sha256":"8ad75ab93a30cf49258c45f8e4ef2bfa346d128857b26a127905a1a906f33dc3","schema_version":"1.0","event_id":"sha256:8ad75ab93a30cf49258c45f8e4ef2bfa346d128857b26a127905a1a906f33dc3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/G7FGUOY4UMKLWTGUMYMMNNGWQM/bundle.json","state_url":"https://pith.science/pith/G7FGUOY4UMKLWTGUMYMMNNGWQM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/G7FGUOY4UMKLWTGUMYMMNNGWQM/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":"2026-06-08T21:35:37Z","links":{"resolver":"https://pith.science/pith/G7FGUOY4UMKLWTGUMYMMNNGWQM","bundle":"https://pith.science/pith/G7FGUOY4UMKLWTGUMYMMNNGWQM/bundle.json","state":"https://pith.science/pith/G7FGUOY4UMKLWTGUMYMMNNGWQM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/G7FGUOY4UMKLWTGUMYMMNNGWQM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:G7FGUOY4UMKLWTGUMYMMNNGWQM","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":"8719a307aaaf9127249a9edb1e8092995349d995dcfc74775678723aacb0c8d1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-09-11T13:31:31Z","title_canon_sha256":"3f43f39b88c06c94e89f3096e81b8fc013377d5414595b712fcfc564ae186ffc"},"schema_version":"1.0","source":{"id":"1409.3430","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.3430","created_at":"2026-05-18T02:43:01Z"},{"alias_kind":"arxiv_version","alias_value":"1409.3430v1","created_at":"2026-05-18T02:43:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.3430","created_at":"2026-05-18T02:43:01Z"},{"alias_kind":"pith_short_12","alias_value":"G7FGUOY4UMKL","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"G7FGUOY4UMKLWTGU","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"G7FGUOY4","created_at":"2026-05-18T12:28:28Z"}],"graph_snapshots":[{"event_id":"sha256:8ad75ab93a30cf49258c45f8e4ef2bfa346d128857b26a127905a1a906f33dc3","target":"graph","created_at":"2026-05-18T02:43:01Z","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 paper we study the problems of invariant and ergodic measures under G-expectation framework. In particular, the stochastic differential equations driven by G-Brownian motion have the unique invariant and ergodic measures. Moreover, the invariant and ergodic measures of G-SDEs are also sublinear expectations. However, the invariant measures may not coincide with ergodic measures, which is different from the classical case.","authors_text":"Falei Wang, Guoqiang Zheng, Hanwu Li, Mingshang Hu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-09-11T13:31:31Z","title":"Invariant and ergodic measures for G-diffusion processes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.3430","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":"sha256:261da63868100b39a0e4ea8b691b11b8297dd4d1041dd7ea36d94f0133f5194b","target":"record","created_at":"2026-05-18T02:43:01Z","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":"8719a307aaaf9127249a9edb1e8092995349d995dcfc74775678723aacb0c8d1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-09-11T13:31:31Z","title_canon_sha256":"3f43f39b88c06c94e89f3096e81b8fc013377d5414595b712fcfc564ae186ffc"},"schema_version":"1.0","source":{"id":"1409.3430","kind":"arxiv","version":1}},"canonical_sha256":"37ca6a3b1ca314bb4cd46618c6b4d6830c1f7d550c7699d694043a1eab6a1945","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"37ca6a3b1ca314bb4cd46618c6b4d6830c1f7d550c7699d694043a1eab6a1945","first_computed_at":"2026-05-18T02:43:01.396282Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:43:01.396282Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bKOUC2CGoesQMClF8qmxw07AWI4P+Hnw9yzEQFNNSrLNcgQ6fF1Uxc2wgQazZVfpynLJW/e1mPUUJCvGoS5HDw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:43:01.396910Z","signed_message":"canonical_sha256_bytes"},"source_id":"1409.3430","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:261da63868100b39a0e4ea8b691b11b8297dd4d1041dd7ea36d94f0133f5194b","sha256:8ad75ab93a30cf49258c45f8e4ef2bfa346d128857b26a127905a1a906f33dc3"],"state_sha256":"62eee361d5ec6279e04b09247baab1ca581f56bacddb55c7b1850b8679fac2bb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QRgSpVI+5W3b71vP9o3344VStOVYxXLuuGj62zVTRqm/NLtp1BI27oHjBJskA42tyh/n25X1Nwl7uHACBMJvBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T21:35:37.685413Z","bundle_sha256":"d7762f36368814b00636efa4c10b9da366455239524fe46927cb61642daa27f6"}}