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