{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:OECQ2A7H2HUZ24E76FTONYP2UI","short_pith_number":"pith:OECQ2A7H","schema_version":"1.0","canonical_sha256":"71050d03e7d1e99d709ff166e6e1faa23436dbcee05b15e63dcabf433a907074","source":{"kind":"arxiv","id":"1106.5637","version":2},"attestation_state":"computed","paper":{"title":"The It\\^o exponential on Lie Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Sim\\~ao N. Stelmastchuk","submitted_at":"2011-06-28T12:09:13Z","abstract_excerpt":"Let $G$ be a Lie Group with a left invariant connection $\\nabla^{G}$. Denote by $\\g$ the Lie algebra of $G$, which is equipped with a connection $\\nabla^{\\g}$. Our main is to introduce the concept of the It\\^o exponential and the It\\^o logarithm, which take in account the geometry of the Lie group $G$ and the Lie algebra $\\g$. This definition characterize directly the martingales in $G$ with respect to the left invariant connection $\\nabla^{G}$. Further, if any $\\nabla^{\\g}$ geodesic in $\\g$ is send in a $\\nabla^{G}$ geodesic we can show that the It\\^o exponential and the It\\^o logarithm are t"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1106.5637","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-06-28T12:09:13Z","cross_cats_sorted":[],"title_canon_sha256":"eb396120f741cde2a90b91d204c100f307a8c027bf33f90c0f16048eb2c8e8d3","abstract_canon_sha256":"8a34789aafcfdf7e7d3232e78a2be341cee14fc1df6c17b7ec5af01db56aac50"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:25:04.415912Z","signature_b64":"9zCNpiYf6cVCKfC5ftsSRWecBYnnSlbuuJ4+xTgViDXySKbw/3WuCQtKw1z9wmL0Oiyhf1VSF/3VapE4S30pBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"71050d03e7d1e99d709ff166e6e1faa23436dbcee05b15e63dcabf433a907074","last_reissued_at":"2026-05-18T03:25:04.415401Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:25:04.415401Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The It\\^o exponential on Lie Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Sim\\~ao N. Stelmastchuk","submitted_at":"2011-06-28T12:09:13Z","abstract_excerpt":"Let $G$ be a Lie Group with a left invariant connection $\\nabla^{G}$. Denote by $\\g$ the Lie algebra of $G$, which is equipped with a connection $\\nabla^{\\g}$. Our main is to introduce the concept of the It\\^o exponential and the It\\^o logarithm, which take in account the geometry of the Lie group $G$ and the Lie algebra $\\g$. This definition characterize directly the martingales in $G$ with respect to the left invariant connection $\\nabla^{G}$. Further, if any $\\nabla^{\\g}$ geodesic in $\\g$ is send in a $\\nabla^{G}$ geodesic we can show that the It\\^o exponential and the It\\^o logarithm are t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.5637","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":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1106.5637","created_at":"2026-05-18T03:25:04.415466+00:00"},{"alias_kind":"arxiv_version","alias_value":"1106.5637v2","created_at":"2026-05-18T03:25:04.415466+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.5637","created_at":"2026-05-18T03:25:04.415466+00:00"},{"alias_kind":"pith_short_12","alias_value":"OECQ2A7H2HUZ","created_at":"2026-05-18T12:26:37.096874+00:00"},{"alias_kind":"pith_short_16","alias_value":"OECQ2A7H2HUZ24E7","created_at":"2026-05-18T12:26:37.096874+00:00"},{"alias_kind":"pith_short_8","alias_value":"OECQ2A7H","created_at":"2026-05-18T12:26:37.096874+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/OECQ2A7H2HUZ24E76FTONYP2UI","json":"https://pith.science/pith/OECQ2A7H2HUZ24E76FTONYP2UI.json","graph_json":"https://pith.science/api/pith-number/OECQ2A7H2HUZ24E76FTONYP2UI/graph.json","events_json":"https://pith.science/api/pith-number/OECQ2A7H2HUZ24E76FTONYP2UI/events.json","paper":"https://pith.science/paper/OECQ2A7H"},"agent_actions":{"view_html":"https://pith.science/pith/OECQ2A7H2HUZ24E76FTONYP2UI","download_json":"https://pith.science/pith/OECQ2A7H2HUZ24E76FTONYP2UI.json","view_paper":"https://pith.science/paper/OECQ2A7H","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1106.5637&json=true","fetch_graph":"https://pith.science/api/pith-number/OECQ2A7H2HUZ24E76FTONYP2UI/graph.json","fetch_events":"https://pith.science/api/pith-number/OECQ2A7H2HUZ24E76FTONYP2UI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OECQ2A7H2HUZ24E76FTONYP2UI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OECQ2A7H2HUZ24E76FTONYP2UI/action/storage_attestation","attest_author":"https://pith.science/pith/OECQ2A7H2HUZ24E76FTONYP2UI/action/author_attestation","sign_citation":"https://pith.science/pith/OECQ2A7H2HUZ24E76FTONYP2UI/action/citation_signature","submit_replication":"https://pith.science/pith/OECQ2A7H2HUZ24E76FTONYP2UI/action/replication_record"}},"created_at":"2026-05-18T03:25:04.415466+00:00","updated_at":"2026-05-18T03:25:04.415466+00:00"}