{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:OECQ2A7H2HUZ24E76FTONYP2UI","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":"8a34789aafcfdf7e7d3232e78a2be341cee14fc1df6c17b7ec5af01db56aac50","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-06-28T12:09:13Z","title_canon_sha256":"eb396120f741cde2a90b91d204c100f307a8c027bf33f90c0f16048eb2c8e8d3"},"schema_version":"1.0","source":{"id":"1106.5637","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1106.5637","created_at":"2026-05-18T03:25:04Z"},{"alias_kind":"arxiv_version","alias_value":"1106.5637v2","created_at":"2026-05-18T03:25:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.5637","created_at":"2026-05-18T03:25:04Z"},{"alias_kind":"pith_short_12","alias_value":"OECQ2A7H2HUZ","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"OECQ2A7H2HUZ24E7","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"OECQ2A7H","created_at":"2026-05-18T12:26:37Z"}],"graph_snapshots":[{"event_id":"sha256:468f65ca589660bfe6ecb164cfba104ac65e90b1d1ccb238fe6e3916d2f2d763","target":"graph","created_at":"2026-05-18T03:25:04Z","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":"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","authors_text":"Sim\\~ao N. Stelmastchuk","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-06-28T12:09:13Z","title":"The It\\^o exponential on Lie Groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.5637","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":"sha256:75ddfc476ce4023f8ce3b8c620ca1cd7b10ab15670123a2363de7fb15ae5b135","target":"record","created_at":"2026-05-18T03:25:04Z","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":"8a34789aafcfdf7e7d3232e78a2be341cee14fc1df6c17b7ec5af01db56aac50","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-06-28T12:09:13Z","title_canon_sha256":"eb396120f741cde2a90b91d204c100f307a8c027bf33f90c0f16048eb2c8e8d3"},"schema_version":"1.0","source":{"id":"1106.5637","kind":"arxiv","version":2}},"canonical_sha256":"71050d03e7d1e99d709ff166e6e1faa23436dbcee05b15e63dcabf433a907074","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"71050d03e7d1e99d709ff166e6e1faa23436dbcee05b15e63dcabf433a907074","first_computed_at":"2026-05-18T03:25:04.415401Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:25:04.415401Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9zCNpiYf6cVCKfC5ftsSRWecBYnnSlbuuJ4+xTgViDXySKbw/3WuCQtKw1z9wmL0Oiyhf1VSF/3VapE4S30pBA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:25:04.415912Z","signed_message":"canonical_sha256_bytes"},"source_id":"1106.5637","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:75ddfc476ce4023f8ce3b8c620ca1cd7b10ab15670123a2363de7fb15ae5b135","sha256:468f65ca589660bfe6ecb164cfba104ac65e90b1d1ccb238fe6e3916d2f2d763"],"state_sha256":"2969f87f7bef4862ec403993048f3da0b7a85da640a9729ac44e37dbe0b5c0e8"}