Electronic Communications in Probability

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

Maher Boudabra and Greg Markowsky

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Abstract

In [7] it was proved that, given a distribution $\mu $ with zero mean and finite second moment, there exists a simply connected domain $\Omega $ such that if $Z_{t}$ is a standard planar Brownian motion, then $\mathcal{R} e(Z_{\tau _{\Omega }})$ has the distribution $\mu $, where $\tau _{\Omega }$ denotes the exit time of $Z_{t}$ from $\Omega $. In this note, we extend this method to prove that if $\mu $ has a finite $p$-th moment then the first exit time $\tau _{\Omega }$ from $\Omega $ has a finite moment of order $\frac{p} {2}$. We also prove a uniqueness principle for this construction, and use it to give several examples.

Article information

Source
Electron. Commun. Probab., Volume 25 (2020), paper no. 20, 13 pp.

Dates
Received: 27 November 2019
Accepted: 24 February 2020
First available in Project Euclid: 29 February 2020

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1582945213

Digital Object Identifier
doi:10.1214/20-ECP300

Subjects
Primary: 60J65: Brownian motion [See also 58J65] 30C20: Conformal mappings of special domains

Keywords
Planar Brownian motion conformal invariance

Rights
Creative Commons Attribution 4.0 International License.

Citation

Boudabra, Maher; Markowsky, Greg. Remarks on Gross’ technique for obtaining a conformal Skorohod embedding of planar Brownian motion. Electron. Commun. Probab. 25 (2020), paper no. 20, 13 pp. doi:10.1214/20-ECP300. https://projecteuclid.org/euclid.ecp/1582945213


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