Distortion of area and conditioned Brownian motion
type: Item
instance of: scholarly article
title: Distortion of area and conditioned Brownian motion
zbMATH Open document ID: 0794.60081
DOI: 10.1007/BF01292679
author: Philip S. Griffin
author: Gregory Verchota
author: Andrew Vogel
publication date: 1994-08-15T00:00:00Z
review text: In a simply connected planar domain \(D\) the expected lifetime of conditioned Brownian motion may be viewed as a function on the set of hyperbolic geodesics for the domain. We show that each hyperbolic geodesic \(\gamma\) induces a decomposition of \(D\) into disjoint subregions \(\Omega_ j\), \(\bigcup_ j \Omega_ j=D\), and that the subregions are obtained in a natural way by using Euclidean geometric quantities relating \(\gamma\) to \(D\). The lifetime associated with \(\gamma\) on each \(\Omega_ j\) is then shown to be bounded by the product of the diameter of the smallest ball containing \(\gamma \cap \Omega_ j\) and the diameter of the largest ball in \(\Omega_ j\). Because this quantity is never larger than, and in general is much smaller than, the area of the largest ball in \(\Omega_ j\), it leads to finite lifetime estimates in a variety of domains of infinite area.
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 31A15
zbMATH DE Number: 569084
zbMATH Keywords: Brownian motion
zbMATH Keywords: hyperbolic geodesic
zbMATH Keywords: Euclidean geometric quantities
zbMATH Keywords: finite lifetime estimates
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 569084