### 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 Ω_{j} is then shown to be bounded by the product of the diameter of the smallest ball containing γ{n-ary intersection}Ω_{j} and the diameter of the largest ball in Ω_{j}. Because this quantity is never larger than, and in general is much smaller than, the area of the largest ball in Ω_{j} it leads to finite lifetime estimates in a variety of domains of infinite area.

Original language | English (US) |
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Pages (from-to) | 385-413 |

Number of pages | 29 |

Journal | Probability Theory and Related Fields |

Volume | 96 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 1993 |

### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty