Summary
In a simply connected planar domainD 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 ofD into disjoint subregions\(\Omega _j \mathop \cup \limits_j \Omega _j = D\) and that the subregions are obtained in a natural way using Euclidean geometric quantities relating γ toD. 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⋂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.
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Research of the first author was supported in part by NSF Grant DMS-9100811
Research of the second author was supported in part by NSF Grant DMS-9105407
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Griffin, P.S., Verchota, G.C. & Vogel, A.L. Distoriton of area and conditioned Brownian motion. Probab. Th. Rel. Fields 96, 385–413 (1993). https://doi.org/10.1007/BF01292679
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DOI: https://doi.org/10.1007/BF01292679
Mathematics Subject Classification (1991)
- 60J65
- 31A15