Distortion 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 \(\gamma\) 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.
scientific article; zbMATH DE number 569084
scientific article