{"paper":{"title":"Bounding Ornstein-Uhlenbeck Processes and Alikes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jian-Sheng Xie","submitted_at":"2014-07-10T08:54:40Z","abstract_excerpt":"In this note we consider SDEs of the type $\\mathrm{d} X_t=[F (X_t) -A X_t] \\mathrm{d} t +D \\mathrm{d} W_t$ under the assumptions that $A$'s eigenvalues are all of positive real parts and $F (\\cdot)$ has slower-than-linear growth rate. It is proved that $\\displaystyle \\varlimsup_{t \\to \\infty} \\frac{\\|X_t\\|}{\\sqrt{\\log t}} =\\sqrt{2 \\lambda_1}$ almost surely with $\\lambda_1$ being the largest eigenvalue of the matrix $\\displaystyle \\Sigma :=\\int_0^\\infty e^{-s A} \\cdot (D \\cdot D^T) \\cdot e^{-s A^T} \\mathrm{d} s$; the discarded measure-zero set can be chosen independent of the initial values $X_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.2725","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"}