A Collection of Results Relating the Geometry of Plane Domains and the Exit Time of Planar Brownian Motion

Abstract We prove a number of results relating exit times planar Brownian motion with the geometric properties domains in question. Included are proofs conformal invariance moduli rectangles and annuli using motion; similarly probabilistic some recent Karafyllia on harmonic measure starlike domains; examples their complements which simultaneously large when measured by moments time motion, small; several identities involving Cauchy distribution optional stopping theorem.