# Remarks on Gross' technique for obtaining a conformal Skorohod embedding of planar Brownian motion

## TL;DR

This paper extends Gross's technique for conformal Skorohod embedding of planar Brownian motion, demonstrating that finite $L^p$ moments of the target distribution imply finite moments of the exit time, enhancing understanding of embedding properties.

## Contribution

It generalizes Gross's method by establishing a link between the $L^p$ moments of the target distribution and the moments of the exit time in the conformal embedding.

## Key findings

- Finite $L^p$ moments of the distribution imply finite moments of the exit time.
- Extension of Gross's technique to broader class of distributions.
- Provides new insights into the moments of exit times in conformal embeddings.

## Abstract

In a recent work by Gross, it was proved that, given a distribution $\mu$ with zero mean and finite second moment, we can find a simply connected domain $\Omega$ such that if $Z_{t}$ is a standard planar BM, then $\mathcal{R}e(Z_{\tau_{\Omega}})$ has the distribution $\mu$. In this note, we extend his method to prove that if $\mu$ has a finite $L^{p}$ moment then the exit time $\tau_{\Omega}$ has a finite moment of order $\frac{p}{2}$.