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.
"A conformal Skorokhod embedding." Electron. Commun. Probab. 24 1 - 11, 2019. https://doi.org/10.1214/19-ECP272