Extremal problems for conditioned brownian motion and the hyperbolic metric

Rodrigo Bañuelos; Tom Carroll

Annales de l'institut Fourier (2000)

  • Volume: 50, Issue: 5, page 1507-1532
  • ISSN: 0373-0956

Abstract

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This paper investigates isoperimetric-type inequalities for conditioned Brownian motion and their generalizations in terms of the hyperbolic metric. In particular, a generalization of an inequality of P. Griffin, T. McConnell and G. Verchota, concerning extremals for the lifetime of conditioned Brownian motion in simply connected domains, is proved. The corresponding lower bound inequality is formulated in various equivalent forms and a special case of these is proved.

How to cite

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Bañuelos, Rodrigo, and Carroll, Tom. "Extremal problems for conditioned brownian motion and the hyperbolic metric." Annales de l'institut Fourier 50.5 (2000): 1507-1532. <http://eudml.org/doc/75462>.

@article{Bañuelos2000,
abstract = {This paper investigates isoperimetric-type inequalities for conditioned Brownian motion and their generalizations in terms of the hyperbolic metric. In particular, a generalization of an inequality of P. Griffin, T. McConnell and G. Verchota, concerning extremals for the lifetime of conditioned Brownian motion in simply connected domains, is proved. The corresponding lower bound inequality is formulated in various equivalent forms and a special case of these is proved.},
author = {Bañuelos, Rodrigo, Carroll, Tom},
journal = {Annales de l'institut Fourier},
keywords = {hyperbolic geodesics; conditioned Brownian motion},
language = {eng},
number = {5},
pages = {1507-1532},
publisher = {Association des Annales de l'Institut Fourier},
title = {Extremal problems for conditioned brownian motion and the hyperbolic metric},
url = {http://eudml.org/doc/75462},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Bañuelos, Rodrigo
AU - Carroll, Tom
TI - Extremal problems for conditioned brownian motion and the hyperbolic metric
JO - Annales de l'institut Fourier
PY - 2000
PB - Association des Annales de l'Institut Fourier
VL - 50
IS - 5
SP - 1507
EP - 1532
AB - This paper investigates isoperimetric-type inequalities for conditioned Brownian motion and their generalizations in terms of the hyperbolic metric. In particular, a generalization of an inequality of P. Griffin, T. McConnell and G. Verchota, concerning extremals for the lifetime of conditioned Brownian motion in simply connected domains, is proved. The corresponding lower bound inequality is formulated in various equivalent forms and a special case of these is proved.
LA - eng
KW - hyperbolic geodesics; conditioned Brownian motion
UR - http://eudml.org/doc/75462
ER -

References

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  1. [1] L. AHLFORS, Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill, New York, 1973. Zbl0272.30012MR50 #10211
  2. [2] C. BAN DLE, Isoperimetric Inequalities and Applications, Pitman, London, 1980. Zbl0436.35063MR81e:35095
  3. [3] R. BAÑUELOS and T. CARROLL, Conditioned Brownian Motion and Hyperbolic Geodesics in Simply Connected Domains, Michigan Math. J., 40 (1993), 321-332. Zbl0805.60077MR94k:60116
  4. [4] R. BAÑUELOS and E. HOUSWORTH, An Isoperimetric-type Inequality for Integrals of Green's Functions, Michigan Math. J., 42 (1995), 603-611. Zbl0849.31006MR96j:30038
  5. [5] A.F. BEARDON, The Geometry of Discrete Groups, Graduate Texts in Mathematics 91, Springer-Verlag, New York, 1983. Zbl0528.30001MR85d:22026
  6. [6] M. CRANSTON and T.R. MCCONNELL, The Lifetime of Conditioned Brownian Motion, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 65 (1983), 1-11. Zbl0506.60071MR85d:60150
  7. [7] P.S. GRIFFIN, T.R. MCCONNELL and G.C. VERCHOTA, Conditioned Brownian Motion in Simply Connected Planar Domains, Ann. Inst. Henri Poincaré, 29 (1993), 229-249. Zbl0777.60073MR94g:60155
  8. [8] P.S. GRIFFIN, G.C. VERCHOTA and A.L. VOGEL, Distortion of Area and Conditioned Brownian Motion, Probab. Theory Relat. Fields, 96 (1993), 385-413. Zbl0794.60081MR94j:60149
  9. [9] Y. KATZNELSON, An Introduction to Harmonic Analysis, Wiley, New York, 1968. Zbl0169.17902MR40 #1734
  10. [10] S.G. KRANTZ, Complex Analysis: the Geometric Viewpoint, Carus Mathematical Monographs 23, Mathematical Association of America, 1990. Zbl0743.30002MR92a:30026
  11. [11] Ch. POMMERENKE, Univalent Functions, Vandenhoeck and Ruprecht, Gottingen, 1975. Zbl0298.30014
  12. [12] J. XU, The Lifetime of Conditioned Brownian Motion in Planar Domains of Infinite Area, Probab. Theory Relat. Fields, 87 (1991), 469-487. Zbl0718.60092MR92d:60086

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