59 research outputs found
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 -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
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