{"paper":{"title":"First passage in an interval for fractional Brownian motion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cond-mat.stat-mech","authors_text":"Kay Joerg Wiese","submitted_at":"2018-07-23T19:53:30Z","abstract_excerpt":"Be $X_t$ a random process starting at $x \\in [0,1]$ with absorbing boundary conditions at both ends of the interval. Denote $P_1(x)$ the probability to first exit at the upper boundary. For Brownian motion, $P_1(x)=x$, equivalent to $P_1'(x)=1$. For fractional Brownian motion with Hurst exponent $H$, we establish that $P_1'(x) = {\\cal N} [x(1-x)]^{\\frac1H -2} e^{\\epsilon {\\cal F}(x)+ {\\cal O}(\\epsilon^2)}$, where $\\epsilon=H-\\frac12$. The function ${\\cal F}(x)$ is analytic, and well approximated by its Taylor expansion, ${\\cal F}(x)\\simeq 16 (C-1) (x-1/2)^2 +{\\cal O}(x-1/2)^4$, where $C= 0.915"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.08807","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"}