  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.
