Remarks on gross’ technique for obtaining a conformal skorohod embedding of planar brownian motion

In [7] it was proved that, given a distribution µ with zero mean and finite second moment, there exists a simply connected domain Ω such that if Z_t is a standard planar Brownian motion, then Re(Z_TΩ) has the distributionµ, where T Ω denotes the exit time of Z_t from Ω. In this note, we extend this method to prove that if µ has a finite p-th moment then the first exit time T_Ω from Ω has a finite moment of order. We also prove a uniqueness principle for this construction, and use it to give several examples.