Distoriton of area and conditioned Brownian motion

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Abstract

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 γ induces a decomposition of D into disjoint subregions {Mathematical expression} and that the subregions are obtained in a natural way using Euclidean geometric quantities relating γ to D. The lifetime associated with γ on each Ωj is then shown to be bounded by the product of the diameter of the smallest ball containing γ{n-ary intersection}Ωj and the diameter of the largest ball in Ωj. Because this quantity is never larger than, and in general is much smaller than, the area of the largest ball in Ωj it leads to finite lifetime estimates in a variety of domains of infinite area.

Original languageEnglish (US)
Pages (from-to)385-413
Number of pages29
JournalProbability Theory and Related Fields
Volume96
Issue number3
DOIs
StatePublished - Sep 1 1993

Keywords

  • Mathematics Subject Classification (1991): 60J65, 31A15

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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