Conformal Skorokhod embeddings of the uniform distribution and related extremal problems
Phanuel Mariano, Hugo Panzo · arXiv (Cornell University) · 2020
Let $\mu$ be a probability distribution with zero mean and finite nonzero variance. The conformal Skorokhod embedding problem (CSEP) asks for a simply connected domain $D\subset\mathbb{C}$ containing $0$ 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 posed and solved in Gross (2019) where the author gives an explicit construction of a solution domain for any $\mu$. In this paper we give an example of a solution domain $\mathbb{U}$ for the uniform distribution $\mathrm{U}[-1,1]$ which differs from Gross' and possesses the following extremal property: If $D$ is any solution to the CSEP for $\mathrm{U}[-1,1]$, 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 domain to the CSEP for any $\mu$. The proofs rely on a recent spectral upper bound of the torsion function as well as a precise relationship between the widths of the orthogonal projections of a simply connected planar domain and the support of its harmonic measure which is developed in the paper.