Projecting the distribution of planar Browian motion at a stopping time through an analytic function

Greg Markowsky

Published 2016-06-10Version 1

A method is given of deriving the distribution of planar Brownian motion evaluated at certain stopping times using analytic functions. This method relies upon a generalization of the standard conformal invariance of harmonic measure. A number of examples are given, including several in which the stopping time in question is not the exit time of a domain. It is also shown how appropriate choices of domains and stopping times can lead to new proofs of identities, including Euler's Basel sum and the Fourier transform for cosecant.