Advertisement

Exit Times of Brownian Motion, Harmonic Majorization, and Hardy Spaces

  • Burgess DavisEmail author
  • Renming Song
Chapter
Part of the Selected Works in Probability and Statistics book series (SWPS)

Abstract

Let R be an open, connected subset of R n \(n \geqslant 2\), X a Brownian motion in R n starting at a point x in R, and \(\tau\) the first time X leaves R:
$$\tau \left( \omega \right) = \inf \left\{ {t >0:{X_t}(\omega) \notin R} \right\}.$$

Keywords

Brownian Motion Hardy Space Optimal Choice Harmonic Measure Exit Time 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    T. W. ANdersonThe integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities, Proc. Amer. Math. Soc. 6(1955), 170-176.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 21.
    A. BaernsteinIntegral means, univalent functions and circular symmetrization, Acta Math. 133(1974), 139-169.CrossRefMathSciNetGoogle Scholar
  3. 3.
    A. Baernstein and B. A. TaylorSpherical rearrangements, subharmonic functions, and -functions in n-space, Duke Math. J. 43(1976), 245-268.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    C. Borell, Undersökning av paraboliska matt, to appear.Google Scholar
  5. 5.
    D. L. BurkholderDistribution function inequalities for martingales, Ann. Probability 1(1973), 19-42.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    D. L. BurkholderH p spaces and exit times of Brownian motion, Bull. Amer. Math. Soc. 81(1975), 556-558.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    D. L. Burkholder and R. F. GundyExtrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124(1970), 249-304.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    D. L. BurkholderR. F. Gundy, and M. L. SilversteinA maximal function characterization of the class H p Trans. Amer. Math. Soc. 157(1971), 137-153.zbMATHMathSciNetGoogle Scholar
  9. 9.
    J. L. Doob"Stochastic Processes," Wiley, New York, 1953.zbMATHGoogle Scholar
  10. 10.
    J. L. DoobSeminartingales and subharmonic functions, Trans. Amer. Math. Soc. 77(1954), 86-121.zbMATHMathSciNetGoogle Scholar
  11. 11.
    J. L. DoobConformally invariant cluster value theory, Illinois J. Math. 5(1961), 521-549.zbMATHMathSciNetGoogle Scholar
  12. 12.
    P. L. Duren"Theory of H pSpaces," Academic Press, New York, 1970.Google Scholar
  13. 13.
    E. B. Dynkin and A. A. Yushkevich"Markov Processes: Theorems and Problems," Plenum, New York, 1969.Google Scholar
  14. 14.
    A. ErdélyiW. MagnusF. Oberhettinger, and F. G. Tricomi"Higher Transcendental Functions," Vol. I, McGraw-Hill, New York, 1953.Google Scholar
  15. 15.
    K. HalisteEstimates of harmonic measures, Ark. Mat.6 (1965), 1-31.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    L. J. HansenHardy classes and ranges of functions, Michigan Math. J. 17(1970), 235-248.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    L. J. HansenBoundary values and mapping properties of Hpfunctions, Math. Z. 128(1972), 189-194.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    L. L. Helms"Introduction to Potential Theory," Wiley-Interscience, New York, 1969.zbMATHGoogle Scholar
  19. 19.
    H. Helson and D. SarasonPast and future, Math. Scand. 21(1967), 5-16.MathSciNetGoogle Scholar
  20. 20.
    G. A. HuntSome theorems concerning Brownian motion, Trans. Amer. Math. Soc. 81(1956), 294-319.zbMATHGoogle Scholar
  21. 21.
    H. P. McKean"Stochastic Integrals," Academic Press, New York, 1969.zbMATHGoogle Scholar
  22. 22.
    P. A. Meyer"Probability and Potentials," Blaisdell, Waltham, Mass. 1966.zbMATHGoogle Scholar
  23. 23.
    J. Neuwirth and D. J. NewmanPositive H 1/2functions are constants, Proc. Amer. Math. Soc. 18(1967), 958.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Department of StatisticsPurdue UniversityWest LafayetteUSA
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA

Personalised recommendations