{"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"}