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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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

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

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

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

Planar Brownian motion conformal invariance

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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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  • [1] R. Bañuelos and T. Carroll. Brownian motion and the fundamental frequency of a drum. Duke Mathematical Journal, 75(3):575–602, 1994.
  • [2] D. Burkholder. Exit times of Brownian motion, harmonic majorization, and Hardy spaces. Advances in Mathematics, 26(2):182–205, 1977.
  • [3] P. Butzer and R. Nessel. Hilbert transforms of periodic functions. In Fourier Analysis and Approximation, pages 334–354. Springer, 1971.
  • [4] W. Chin, P. Jung, and G. Markowsky. Some remarks on invariant maps of the Cauchy distribution. Statistics and Probability Letters, 158, 2020.
  • [5] W. Feller. An introduction to probability theory and its applications. 1957, 2.
  • [6] L. Grafakos. Classical Fourier analysis, volume 2. Springer, 2008.
  • [7] R. Gross. A conformal Skorokhod embedding. Electronic Communications in Probability, 24(68):1–11, 2019.
  • [8] L. Hansen. Hardy classes and ranges of functions. The Michigan Mathematical Journal, 17(3):235–248, 1970.
  • [9] S. Kanas and T. Sugawa. On conformal representations of the interior of an ellipse. 31(2):329, 2006.
  • [10] F. King. Hilbert transforms. Cambridge University Press Cambridge, 2009.
  • [11] P. Mariano and H. Panzo. Conformal skorokhod embeddings of the uniform distribution and related extremal problems. arXiv:2001.12008, 2020.
  • [12] G. Markowsky. The exit time of planar Brownian motion and the Phragmén–Lindelöf principle. Journal of Mathematical Analysis and Applications, 422(1):638–645, 2015.
  • [13] G. Markowsky. On the distribution of planar Brownian motion at stopping times. Annales Academiæ Scientiarum Fennicæ Mathematica, 2018.
  • [14] R. Remmert. Theory of complex functions, volume 122. Springer Science & Business Media, 2012.
  • [15] W. Rudin. Real and complex analysis. Tata McGraw-Hill, 2006.
  • [16] D. Williams. Probability with martingales. Cambridge university press, 1991.