{"paper":{"title":"Non-exponential stability and decay rates in nonlinear stochastic difference equation with unbounded noises","license":"","headline":"","cross_cats":["math.DS"],"primary_cat":"math.PR","authors_text":"A. Rodkina, G. Berkolaiko, J.A.D. Appleby","submitted_at":"2006-10-13T03:48:05Z","abstract_excerpt":"We consider stochastic difference equation x_{n+1} = x_n (1 - h f(x_n) + \\sqrt{h} g(x_n) \\xi_{n+1}), where functions f and g are nonlinear and bounded, random variables \\xi_i are independent and h>0 is a nonrandom parameter. We establish results on asymptotic stability and instability of the trivial solution x_n=0. We also show, that for some natural choices of the nonlinearities f and g, the rate of decay of x_n is approximately polynomial: we find \\alpha>0 such that x_n decay faster than n^{-\\alpha+\\epsilon} but slower than n^{-\\alpha-\\epsilon} for any \\epsilon>0. It also turns out that if g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0610425","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"}