{"paper":{"title":"Brownian motion and Random Walk above Quenched Random Wall","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Bastien Mallein, Piotr Mi{\\l}o\\'s","submitted_at":"2015-07-30T16:50:11Z","abstract_excerpt":"We study the persistence exponent for the first passage time of a random walk below the trajectory of another random walk. More precisely, let $\\{B_n\\}$ and $\\{W_n\\}$ be two centered, weakly dependent random walks. We establish that $\\mathbb{P}(\\forall_{n\\leq N} B_n \\geq W_n|W) = N^{-\\gamma + o(1)}$ for a non-random $\\gamma\\geq 1/2$. In the classical setting, $W_n \\equiv 0$, it is well-known that $\\gamma = 1/2$. We prove that for any non-trivial $W$ one has $\\gamma>1/2$ and the exponent $\\gamma$ depends only on $\\text{Var}(B_1)/\\text{Var}(W_1)$.\n  Our result holds also in the continuous settin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08578","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":""},"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"}