{"paper":{"title":"Stochastic Persistence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Michel Benaim","submitted_at":"2018-06-21T23:19:17Z","abstract_excerpt":"Let $(X_t)_{t \\geq 0}$ be a continuous time Markov process on some metric space $M,$ leaving invariant a closed subset $M_0 \\subset M,$ called the {\\em extinction set}. We give general conditions ensuring either \"Stochastic persistence\" (Part I) : Limit points of the occupation measure are invariant probabilities over $M_+ = M \\setminus M_0;$ or \"Extinction\" (Part II) : $X_t \\rightarrow M_0$ a.s. In the persistence case we also discuss conditions ensuring the a.s convergence (respectively exponential convergence in total variation) of the occupation measure (respectively the distribution) of $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.08450","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1806.08450/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}