Distoriton of area and conditioned Brownian motion

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.