A probabilistic proof of the Open Mapping Theorem for analytic functions

The conformal invariance of Brownian motion is used to give a short proof of the Open Mapping Theorem for analytic functions

Pascal's Hexagon Theorem implies a Butterfly Theorem in the Complex Projective Plane

This paper proves a generalization of the Butterfly Theorem, a classical Euclidean result, which is valid in the complex projective plane

On the expected exit time of planar Brownian motion from simply connected domains

This paper presents some results on the expected exit time of Brownian motion from simply connected domains in \CC. We indicate a way in which Brownian motion sees the identity function and the Koebe function as the smallest and largest analytic functions, respectively, in the Schlicht class. We also give a sharpening of a result of McConnell's concerning the moments of exit times of Schlicht domains. We then show how a simple formula for expected exit time can be applied in a series of examples. Included in the examples given are the expected exit times from given points of a cardioid and regular $m$-gon, as well as bounds on the expected exit time of an infinite wedge. We also calculate the expected exit time of an infinite strip, and in the process obtain a probabilistic derivation of Euler's result that \zeta(2)=\sum_{n=1}^\ff \frac{1}{n^2}= \frac{\pi^2}{6}. We conclude by showing how the formula can be applied to some domains which are not simply connected

Simple random walk on distance-regular graphs

A survey is presented of known results concerning simple random walk on the class of distance-regular graphs. One of the highlights is that electric resistance and hitting times between points can be explicitly calculated and given strong bounds for, which leads in turn to bounds on cover times, mixing times, etc. Also discussed are harmonic functions, moments of hitting and cover times, the Green's function, and the cutoff phenomenon. The main goal of the paper is to present these graphs as a natural setting in which to study simple random walk, and to stimulate further research in the field