{"paper":{"title":"A Lower Bound for the First Passage Time Density of the Suprathreshold Ornstein-Uhlenbeck Process","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["q-bio.NC"],"primary_cat":"math.PR","authors_text":"Peter J. Thomas","submitted_at":"2011-01-20T14:36:11Z","abstract_excerpt":"We prove that the first passage time density $\\rho(t)$ for an Ornstein-Uhlenbeck process $X(t)$ obeying $dX=-\\beta X dt + \\sigma dW$ to reach a fixed threshold $\\theta$ from a suprathreshold initial condition $x_0>\\theta>0$ has a lower bound of the form $\\rho(t)>k \\exp\\left[-p e^{6\\beta t}\\right]$ for positive constants $k$ and $p$ for times $t$ exceeding some positive value $u$. We obtain explicit expressions for $k, p$ and $u$ in terms of $\\beta$, $\\sigma$, $x_0$ and $\\theta$, and discuss application of the results to the synchronization of periodically forced stochastic leaky integrate-and-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.3915","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"}