Brownian Convergence of Planar Domains and Stability of the Planar Skorokhod Embedding Problem

We present a numerical framework for approximating the $\mu$-domain in the planar Skorokhod embedding problem PSEP, recently introduced in \cite{gross2019}. We show that under weak convergence of a sequence of probability measures $(\mu_{n})_{n}$, the corresponding sequence of $\mu_{n}$-domains converges, in an appropriate sense, to the domain associated with the limit measure $\mu$. In addition, we provide implementation strategies, convergence rate estimates, and a numerical example. The method is robust and versatile, offering a concrete computational approach for the approximation of $\mu$-domains. As part of this analysis, we introduce a novel mode of convergence for planar domains via planar Brownian motion, which we call $p$-Brownian convergence.