On the Dirichlet eigenvalue problem and the conformal Skorokhod embedding problem

TL;DR

This paper introduces a new method for solving the conformal Skorokhod embedding problem, providing minimal spectral rate domains for given measures, and explores their uniqueness and boundary characterization.

Contribution

It offers a novel solution to Gross's problem, generalizes previous examples, and establishes minimal spectral rate domains with uniqueness under certain conditions.

Findings

The new domain construction always has minimal spectral rate.

The domain is unique under specific conditions.

The method allows boundary curve identification.

Abstract

In a recent work by Gross, the following problem was stated and solved: given a measure $\mu$ with finite second moment, find a simply connected domain $U$ in $\CC$ such that the real part of a Brownian motion stopped when it leaves $U$ is distributed as $\mu$. The construction developed by Gross yields a domain which is symmetric with respect to the real axis, but it has been noted by other authors that other domains are also possible, in particular there are a number of examples which have the property that a vertical ray starting at a point in the domain lies entirely within the domain. In this paper we give a new solution to the problem posed by Gross, and show that these other cases noted before are special cases of this method. We further show that the domain generated by this method has the property that it always has the minimal rate (as defined in terms of the spectrum of the Laplacian operator) among all possible domains corresponding to a fixed distribution $\mu$, which gives a partial solution to a question posed by Mariano and Panzo. We show that the domain is unique, provided certain conditions are imposed, and use this to give several examples. We also describe a method for identifying the boundary curve of the domain, and discuss several other related topics.