Electronic Journal of Probability

The Exit Place of Brownian Motion in the Complement of a Horn

Dante DeBlassie

Full-text: Open access

Abstract

Consider the domain lying outside a horn. We determine asymptotics of the logarithm of the chance that Brownian motion in the domain has a large exit place. For a certain class of horns, the behavior is given explicitly in terms of the geometry of the domain. We show that for some horns the behavior depends on the dimension, whereas for other horns, it does not. Analytically, the result is equivalent to estimating the harmonic measure of the part of the domain lying outside a cylinder with large diameter.

Article information

Source
Electron. J. Probab., Volume 13 (2008), paper no. 36, 1068-1095.

Dates
Accepted: 9 July 2008
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819109

Digital Object Identifier
doi:10.1214/EJP.v13-524

Mathematical Reviews number (MathSciNet)
MR2424987

Zentralblatt MATH identifier
1190.60076

Subjects
Primary: 60J65: Brownian motion [See also 58J65]

Keywords
Horn-shaped domain $h$-transform Feynman-Kac representation exit place of Brownian motion harmonic measure

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

DeBlassie, Dante. The Exit Place of Brownian Motion in the Complement of a Horn. Electron. J. Probab. 13 (2008), paper no. 36, 1068--1095. doi:10.1214/EJP.v13-524. https://projecteuclid.org/euclid.ejp/1464819109


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References

  • Aikawa, Hiroaki; Murata, Minoru. Generalized Cranston-McConnell inequalities and Martin boundaries of unbounded domains. J. Anal. Math. 69 (1996), 137–152.
  • Bañuelos, Rodrigo; Carroll, Tom. Sharp integrability for Brownian motion in parabola-shaped regions. J. Funct. Anal. 218 (2005), no. 1, 219–253.
  • Bañuelos, Rodrigo; DeBlassie, Dante. The exit distribution of iterated Brownian motion in cones. Stochastic Process. Appl. 116 (2006), no. 1, 36–69.
  • Bass, Richard F. Probabilistic techniques in analysis. Springer-Verlag, New York, 1995.
  • Burkholder, D. L. Exit times of Brownian motion, harmonic majorization, and Hardy spaces. Advances in Math. 26 (1977), no. 2, 182–205.
  • Carroll, Tom; Hayman, Walter K. Conformal mapping of parabola-shaped domains. Comput. Methods Funct. Theory 4 (2004), no. 1, 111–126.
  • Collet, Pierre; Martinez, Servet; San Martin, Jaime. Ratio limit theorem for parabolic horn-shaped domains. Trans. Amer. Math. Soc. 358 (2006), no. 11, 5059–5082 (electronic).
  • DeBlassie, Dante. The change of a long lifetime for Brownian motion in a horn-shaped domain. Electron. Comm. Probab. 12 (2007), 134–139 (electronic).
  • Durrett, Richard. Brownian motion and martingales in analysis. Wadsworth International Group, Belmont, CA, 1984.
  • Essén, Matts; Haliste, Kersti. A problem of Burkholder and the existence of harmonic majoriants of $\vert x\vert \sp{p}$ in certain domains in $R\sp{d}$. Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 107–116.
  • Gilbarg, David; Trudinger, Neil S. Elliptic partial differential equations of second order. Second edition. Springer-Verlag, Berlin, 1983.
  • Haliste, Kersti. Some estimates of harmonic majorants. Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 117–124.
  • Ikeda, Nobuyuki; Watanabe, Shinzo. Stochastic differential equations and diffusion processes. Second edition. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.
  • Ioffe, Dimitry; Pinsky, Ross. Positive harmonic functions vanishing on the boundary for the Laplacian in unbounded horn-shaped domains. Trans. Amer. Math. Soc. 342 (1994), no. 2, 773–791.
  • Itô, Kiyosi; McKean, Henry P., Jr. Diffusion processes and their sample paths. Second printing, corrected. Springer-Verlag, Berlin-New York, 1974.
  • Murata, Minoru. Martin boundaries of elliptic skew products, semismall perturbations, and fundamental solutions of parabolic equations. J. Funct. Anal. 194 (2002), no. 1, 53–141.
  • Murata, Minoru. Uniqueness theorems for parabolic equations and Martin boundaries for elliptic equations in skew product form. J. Math. Soc. Japan 57 (2005), no. 2, 387–413.
  • Pinsky, Ross G. Positive harmonic functions and diffusion. Cambridge University Press, Cambridge, 1995.
  • Stroock, Daniel W.; Varadhan, S. R. Srinivasa. Multidimensional diffusion processes. Springer-Verlag, Berlin-New York, 1979.
  • Warschawski, S. E. On conformal mapping of infinite strips. Trans. Amer. Math. Soc. 51 (1942). 280–335. (4,9b)