{"paper":{"title":"Bounding the Solutions to Some SDEs via Ergodic Theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.PR","authors_text":"Jian-Sheng Xie","submitted_at":"2014-07-10T09:05:54Z","abstract_excerpt":"In this note we consider autonomous SDEs admitting smooth invariant measures. We present a method in finding (almost everywhere) good bounds for $\\sup \\{\\|X_t\\|: t \\in [0, T]\\}$ for strong solutions $X_{\\cdot}$ to such SDEs, which in many cases are optimal bounds. In some situation (especially in one-dimensional SDEs' cases), the discarded measure-zero set can be chosen to be a measure-zero set of the underlying Brownian motion uniform for all initial points $X_0=x$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.2728","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"}