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Maximizing the pth moment of the exit time of planar brownian motion from a given domain

Published online by Cambridge University Press:  23 November 2020

Maher Boudabra  and
Greg Markowsky
Maher Boudabra*
Affiliation:
Monash University
Greg Markowsky*
Affiliation:
Monash University
*
*Postal address: Clayton, Victoria, Australia.
*Postal address: Clayton, Victoria, Australia.
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Abstract

In this paper we address the question of finding the point which maximizes the pth moment of the exit time of planar Brownian motion from a given domain. We present a geometrical method for excluding parts of the domain from consideration which makes use of a coupling argument and the conformal invariance of Brownian motion. In many cases the maximizing point can be localized to a relatively small region. Several illustrative examples are presented.

Keywords

Planar Brownian motionexit time
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 4 , December 2020 , pp. 1135 - 1149
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Copyright
© Applied Probability Trust 2020

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References

Bañuelos, R. and Burdzy, K. (1999). On the ‘hot spots’ conjecture of J. Rauch. J. Funct. Anal. 164 (1), 1–33.10.1006/jfan.1999.3397CrossRefGoogle Scholar
Bañuelos, R. and Carroll, T. (1994). Brownian motion and the fundamental frequency of a drum. Duke Math. J. 75 (3), 575–602.Google Scholar
Bañuelos, R. and Carroll, T. (2011). The maximal expected lifetime of Brownian motion. Math. Proc. Roy. Irish Acad. 111A (1), 1–11.10.3318/PRIA.2011.111.1CrossRefGoogle Scholar
Bañuelos, R., Pang, M. and Pascu, M. (2004). Brownian motion with killing and reflection and the ‘hot-spots’ problem. Prob. Theory Relat. Fields 130, 56–68.Google Scholar
Bañuelos, R., Mariano, P. and Wang, J. (2020). Bounds for exit times of Brownian motion and the first Dirichlet eigenvalue for the Laplacian. arXiv:2003.06867.Google Scholar
Bass, R. (1994). Probabilistic Techniques in Analysis. Springer.Google Scholar
Burkholder, D. (1977). Exit times of Brownian motion, harmonic majorization, and Hardy spaces. Adv. Math. 26 (2), 182–205.10.1016/0001-8708(77)90029-9CrossRefGoogle Scholar
Driscoll, T. and Trefethen, L. N. (2002). Schwarz–Christoffel Mapping (Cambridge Monographs on Applied and Computational Mathematics 8). Cambridge University Press.Google Scholar
Keady, G. and McNabb, A. (1993). The elastic torsion problem: solutions in convex domains. New Zealand J. Math. 22, 43–64.Google Scholar
Kern, D. (1950). Process Heat Transfer. McGraw-Hill.Google Scholar
Kim, D. (2019). Quantitative inequalities for the expected lifetime of Brownian motion. To appear in Michigan Math. J. Available at arXiv:1904.09565.Google Scholar
Makar-Limanov, L. (1971). Solution of Dirichlet’s problem for the equation $\bigtriangleup u=-1$ in a convex region. Math. Notes Acad. Sci. USSR 9 (1), 52–53.Google Scholar
Markowsky, G. (2015). The exit time of planar Brownian motion and the Phragmén–Lindelöf principle. J. Math. Anal. Appl. 422 (1), 638–645.Google Scholar
Méndez-Hernández, P. (2002). Brascamp–Lieb–Luttinger inequalities for convex domains of finite inradius. Duke Math. J. 113 (1), 93–131.Google Scholar
Mörters, P. and Peres, Y. (2010). Brownian Motion (Cambridge Series in Statistical and Probabilistic Mathematics). Cambridge University Press.Google Scholar
Pascu, M. (2002). Scaling coupling of reflecting Brownian motions and the hot spots problem. Trans. Amer. Math. Soc. 354 (11), 4681–4702.Google Scholar
Philippin, G. and Porru, G. (1996). Isoperimetric inequalities and overdetermined problems for the Saint-Venant equation. New Zealand J. Math. 25, 217–227.Google Scholar
Sperb, R. P. (ed.) (1981). Maximum Principles and their Applications. Elsevier.Google Scholar