{"paper":{"title":"On certain functionals of the maximum of Brownian motion and their applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.dis-nn","math-ph","math.MP","math.PR"],"primary_cat":"cond-mat.stat-mech","authors_text":"Alain Comtet, Anthony Perret, Gregory Schehr, Satya N. Majumdar","submitted_at":"2015-02-04T14:51:12Z","abstract_excerpt":"We consider a Brownian motion (BM) $x(\\tau)$ and its maximal value $x_{\\max} = \\max_{0 \\leq \\tau \\leq t} x(\\tau)$ on a fixed time interval $[0,t]$. We study functionals of the maximum of the BM, of the form ${\\cal O}_{\\max}(t)=\\int_0^t\\, V(x_{\\max} - x(\\tau)) {\\rm d} \\tau$ where $V(x)$ can be any arbitrary function and develop various analytical tools to compute their statistical properties. These tools rely in particular on (i) a \"counting paths\" method and (ii) a path-integral approach. In particular, we focus on the case where $V(x) = \\delta(x-r)$, with $r$ a real parameter, which is releva"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.01218","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"}