Skip to main content
SpringerLink
Log in

Uniform dimension results for Gaussian random fields

  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

Let X = {X(t), t ∈ ℝN} be a Gaussian random field with values in ℝd defined by

$$ X(t) = (X_1 (t),...,X_d (t)),\forall t \in \mathbb{R}^N . $$
((1))

. The properties of space and time anisotropy of X and their connections to uniform Hausdorff dimension results are discussed. It is shown that in general the uniform Hausdorff dimension result does not hold for the image sets of a space-anisotropic Gaussian random field X.

When X is an (N, d)-Gaussian random field as in (1), where X 1,...,X d are independent copies of a real valued, centered Gaussian random field X 0 which is anisotropic in the time variable. We establish uniform Hausdorff dimension results for the image sets of X. These results extend the corresponding results on one-dimensional Brownian motion, fractional Brownian motion and the Brownian sheet.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Samorodnitsky G, Taqqu M S. Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. New York: Chapman & Hall, 1994

    MATH  Google Scholar 

  2. Adler R J. The Geometry of Random Fields. New York: Wiley, 1981

    MATH  Google Scholar 

  3. Kahane J P. Some Random Series of Functions. 2nd Ed. Cambridge: Cambridge University Press, 1985

    MATH  Google Scholar 

  4. Khoshnevisan D. Multiparameter Processes: An Introduction to Random Fields. New York: Springer, 2002

    MATH  Google Scholar 

  5. Falconer K J. Fractal Geometry — Mathematical Foundations and Applications. Chichester: John Wiley & Sons, 1990

    MATH  Google Scholar 

  6. Kaufman R. Une propriété métrique du mouvement brownien. C R Acad Sci Paris, 268: 727–728 (1968)

    MathSciNet  Google Scholar 

  7. Monrad D, Pitt L D. Local nondeterminism and Hausdorff dimension. In: Progress in Probability and Statistics. Seminar on Stochastic Processes 1986. Cinlar E, Chung K L, Getoor R K, eds. Boston: Birkhäuser, 1987, 163–189

    Google Scholar 

  8. Mountford T S. Uniform dimension results for the Brownian sheet. Ann Probab, 17: 1454–1462 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  9. Khoshnevisan D, Xiao Y. Images of the Brownian sheet. Trans Amer Math Soc, 359: 3125–3151 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Khoshnevisan D, Wu D, Xiao Y. Sectorial local nondeterminism and the geometry of the Brownian sheet. Electron J Probab, 11: 817–843 (2006)

    MathSciNet  Google Scholar 

  11. Davies S, Hall P. Fractal analysis of surface roughness by using spatial data (with discussion). J Royal Statist Soc Ser B, 61: 3–37 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bonami A, Estrade A. Anisotropic analysis of some Gaussian models. J Fourier Anal Appl, 9: 215–236 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Benson D A, Meerschaert M M, Baeumer B, et al. Aquifer operator-scaling and the effect on solute mixing and dispersion. Water Resources Research, 42: W01415 (2008)

    Article  Google Scholar 

  14. Kamont A. On the fractional anisotropic Wiener field. Probab Math Statist, 16: 85–98 (1996)

    MATH  MathSciNet  Google Scholar 

  15. Mueller C, Tribe R. Hitting probabilities of a random string. Electron J Probab, 7: 1–29 (2002)

    MathSciNet  Google Scholar 

  16. Biermé H, Meerschaert M M, Scheffler H P. Operator scaling stable random fields. Stoch Process Appl, 117: 313–332 (2007)

    Article  Google Scholar 

  17. Xiao Y. Dimension results for Gaussian vector fields and index-α stable fields. Ann Probab, 23: 273–291 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  18. Xiao Y. Hausdorff measure of the graph of fractional Brownian motion. Math Proc Cambridge Philos Soc, 122: 565–576 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  19. Mason D J, Xiao Y. Sample path properties of operator self-similar Gaussian random fields. Teor Veroyatnost i Primenen, 46: 94–116 (2001)

    MathSciNet  Google Scholar 

  20. Ayache A, Xiao Y. Asymptotic properties and Hausdorff dimensions of fractional Brownian sheets. J Fourier Anal Appl, 11: 407–439 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Wu D, Xiao Y. Dimensional properties of fractional Brownian motion. Acta Math Sinica, 23: 613–622 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. Wu D, Xiao Y. Geometric properties of the images fractional Brownian sheets. J Fourier Anal Appl, 13: 1–37 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Xiao Y. Sample path properties of anisotropic Gaussian random fields. In: A Minicourse on Stochastic Partial Differential Equations. Khoshnevisan D, Rassoul-Agha F, eds. New York: Springer, 2009, 145–212

    Chapter  Google Scholar 

  24. Xiao Y. Strong local nondeterminism of Gaussian random fields and its applications. In: Asymptotic Theory in Probability and Statistics with Applications. Lai T L, Shao Q M, Qian L, eds. Beijing: Higher Education Press, 2007, 136–176

    Google Scholar 

  25. Ayache A, Wu D, Xiao Y. Joint continuity of the local times of fractional Brownian sheets. Ann Inst H Poincaré Probab Statist, 44: 727–748 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kaufman R. Measures of Hausdorff-type, and Brownian motion. Mathematika, 19: 115–119 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  27. Hawkes J. Measures of Hausdorff type and stable processes. Mathematika, 25: 202–212 (1978)

    MathSciNet  Google Scholar 

  28. Taylor S J, Watson N A. A Hausdorff measure classification of polar sets for the heat equation. Math Proc Cambridge Philos Soc, 97: 325–344 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  29. Testard F. Polarité, points multiples et géométrie de certain processus gaussiens. Toulouse: Publ du Laboratoire de Statistique et Probabilités de l’U.P.S., 1986, 1–86

    Google Scholar 

  30. Xiao Y. Hitting probabilities and polar sets for fractional Brownian motion. Stochast Stochast Rep, 66: 121–151 (1999)

    MATH  Google Scholar 

  31. Kaufman R. Dimensional properties of one-dimensional Brownian motion. Ann Probab, 17: 189–193 (1989)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL, 35899, USA

    DongSheng Wu

  2. Department of Statistics and Probability, Michigan State University, East Lansing, MI, 48824, USA

    YiMin Xiao

  3. College of Mathematics and Computer Science, Anhui Normal University, Wuhu, 241000, China

    YiMin Xiao

Authors
  1. DongSheng Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. YiMin Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to DongSheng Wu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wu, D., Xiao, Y. Uniform dimension results for Gaussian random fields. Sci. China Ser. A-Math. 52, 1478–1496 (2009). https://doi.org/10.1007/s11425-009-0103-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-009-0103-x

Keywords

MSC(2000)

Search

Navigation