{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:USEO4OHXXQT55MD2LHKIXWO64W","short_pith_number":"pith:USEO4OHX","schema_version":"1.0","canonical_sha256":"a488ee38f7bc27deb07a59d48bd9dee5aa485b9a9de54917baa42c679c4f6f47","source":{"kind":"arxiv","id":"1104.1043","version":1},"attestation_state":"computed","paper":{"title":"Hitting spheres on hyperbolic spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Enzo Orsingher, Valentina Cammarota","submitted_at":"2011-04-06T09:30:03Z","abstract_excerpt":"For a hyperbolic Brownian motion on the Poincar\\'e half-plane $\\mathbb{H}^2$, starting from a point of hyperbolic coordinates $z=(\\eta, \\alpha)$ inside a hyperbolic disc $U$ of radius $\\bar{\\eta}$, we obtain the probability of hitting the boundary $\\partial U$ at the point $(\\bar \\eta,\\bar \\alpha)$. For $\\bar{\\eta} \\to \\infty$ we derive the asymptotic Cauchy hitting distribution on $\\partial \\mathbb{H}^2$ and for small values of $\\eta$ and $\\bar \\eta$ we obtain the classical Euclidean Poisson kernel. The exit probabilities $\\mathbb{P}_z\\{T_{\\eta_1}