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Remarks on Gross’ technique for obtaining a conformal Skorohod embedding of planar Brownian motion. (English) Zbl 1434.60224
Summary: In [R. Gross, Electron. Commun. Probab. 24, Paper No. 68, 11 p. (2019; Zbl 1450.60032)] 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.

60J65 Brownian motion
30C20 Conformal mappings of special domains
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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. · Zbl 0817.58046
[2] D. Burkholder. Exit times of Brownian motion, harmonic majorization, and Hardy spaces. Advances in Mathematics, 26(2):182-205, 1977. · Zbl 0372.60112
[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. · Zbl 0077.12201
[6] L. Grafakos. Classical Fourier analysis, volume 2. Springer, 2008. · Zbl 1220.42001
[7] R. Gross. A conformal Skorokhod embedding. Electronic Communications in Probability, 24(68):1-11, 2019. · Zbl 1450.60032
[8] L. Hansen. Hardy classes and ranges of functions. The Michigan Mathematical Journal, 17(3):235-248, 1970. · Zbl 0189.08602
[9] S. Kanas and T. Sugawa. On conformal representations of the interior of an ellipse. 31(2):329, 2006. · Zbl 1098.30011
[10] F. King. Hilbert transforms. Cambridge University Press Cambridge, 2009. · Zbl 1188.44005
[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. · Zbl 1328.60189
[13] G. Markowsky. On the distribution of planar Brownian motion at stopping times. Annales Academiæ Scientiarum Fennicæ Mathematica, 2018. · Zbl 1404.60117
[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. · Zbl 0142.01701
[16] D. Williams. Probability with martingales. Cambridge university press, 1991. · Zbl 0722.60001
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