## Abstract

In a recent work by Gross, the following problem was posed and solved: given a measure μ on R with finite second moment, find a simply connected domain U in C such that the real part of the random variable Z_{τU} has the distribution μ, where Z is a planar Brownian motion and τ_{U} is the exit time from U. 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, following a method due to Mariano and Panzo, 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 μ. This 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.

Original language | English |
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Article number | 124351 |

Number of pages | 14 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 491 |

Issue number | 2 |

DOIs | |

Publication status | Published - 15 Nov 2020 |

## Keywords

- Conformal mapping
- Fourier series
- Planar Brownian motion
- Simply connected domains
- Skorokhod embedding