The paper proves, via conformal invariance of Brownian motion, that the hyperbolic secant law is invariant under g(x)=(2/π)log|sinh(πx/2)|, and gives a general transfer principle for exit distributions of symmetric domains.
A conformal Skorokhod embedding
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abstract
Start a planar Brownian motion and let it run until it hits some given barrier. We show that the barrier may be crafted so that the x coordinate at the hitting time has any prescribed centered distribution with finite variance. This provides a new, complex-analytic proof of the Skorokhod embedding theorem. Our method is constructive and can give an explicit description of the barrier.
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A note on invariance of the Cauchy and related distributions
The paper proves, via conformal invariance of Brownian motion, that the hyperbolic secant law is invariant under g(x)=(2/π)log|sinh(πx/2)|, and gives a general transfer principle for exit distributions of symmetric domains.