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

A collection of results relating the geometry of plane domains and the exit time of planar Brownian motion