Conformal Skorokhod Embeddings of the uniform distribution and related extremal problems

Phanuel Mariano, Hugo Panzo

Published 2020-01-31Version 1

Let $\mu$ be a probability distribution with zero mean and finite variance. The conformal Skorokhod embedding problem (CSEP) asks for a simply connected domain $D\subset\mathbb{C}$ such that the real part of standard complex Brownian motion at its first exit time from $D$ has distribution $\mu$ with the exit time having finite mean. The CSEP was first posed and solved by Gross (2019) where the author gives an explicit construction of a solution. In this paper we give an example of a simply connected domain $\mathbb{U}$ that embeds the uniform distribution on $[-1,1]$ which differs from Gross' solution and possesses a certain extremal property. More precisely, if $D$ is any solution of the CSEP for the uniform distribution on $[-1,1]$, then the principal Dirichlet eigenvalue of $D$ is at least that of $\mathbb{U}$. We also give general upper and lower bounds on the principal Dirichlet eigenvalue of a solution to the CSEP for any $\mu$. The proofs rely on a spectral upper bound of the torsion function as well as a precise relation between the widths of the orthogonal projections of a simply connected domain and the support of its harmonic measure which is developed in the paper.