Advertisement

Hausdorff Measure of the Range of Space–Time Anisotropic Gaussian Random Fields

  • Wenqing Ni
  • Zhenlong ChenEmail author
Article
  • 8 Downloads

Abstract

Let \(X=\{X(t)\in {{\mathbb {R}}}^d, t\in {{\mathbb {R}}}^N\}\) be a centered space–time anisotropic Gaussian random field with stationary increments, whose components are independent but may not be identically distributed. Under certain mild conditions, we determine the exact Hausdorff measure function for the range \(X([0,1]^N)\). Our result extends those in Talagrand (Ann Probab 23:767–775, 1995) for fractional Brownian motion and Luan and Xiao (J Fourier Anal Appl 18:118–145, 2012) for time-anisotropic and space-isotropic Gaussian random fields.

Keywords

Space–time anisotropic Gaussian random fields Strong local nondeterminism Range Hausdorff measure 

Mathematics Subject Classification (2010)

60G15 60G17 60G60 

Notes

References

  1. 1.
    Adler, R.J.: The Geometry of Random Fields. Wiley, New York (1981)zbMATHGoogle Scholar
  2. 2.
    Biermé, H., Lacaux, C., Xiao, Y.: Hitting probabilities and the Hausdorff dimension of the inverse images of anisotropic Gaussian random fields. Bull. London Math. Soc. 41, 253–273 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chen, Z., Xiao, Y.: On intersections of independent anisotropic Gaussian random fields. Sci. China Math. 55, 2217–2232 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Estrade, A., Wu, D., Xiao, Y.: Packing dimension results for anisotropic Gaussian random fields. Commun. Stoch. Anal. 5, 41–64 (2011)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Falconer, K.J.: Fractal Geometry—Mathematical Foundations and Applications, 2nd edn. Wiley, Chichester (2003)CrossRefGoogle Scholar
  6. 6.
    Kahane, J.P.: Some Random Series of Functions, 2nd edn. Cambridge University Press, Cambridge (1985)zbMATHGoogle Scholar
  7. 7.
    Khoshnevisan, D.: Multiparameter Processes: An Introduction to Random Fields. Springer, New York (2002)CrossRefGoogle Scholar
  8. 8.
    Luan, N., Xiao, Y.: Spectral conditions for strong local nondeterminism and exact Hausdorff measure of ranges of Gaussian random fields. J. Fourier Anal. Appl. 18, 118–145 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  10. 10.
    Ni, W., Chen, Z.: Hitting probabilities for a class of Gaussian random fields. Stat. Probab. Lett. 118, 145–155 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ni, W., Chen, Z.: Dimension results and hitting probabilities of space–time anisotropic Gaussian random fields (in Chinese). Sci. Sin. China 48, 1–24 (2018)CrossRefGoogle Scholar
  12. 12.
    Pruitt, W.E., Taylor, S.J.: Sample path properties of processes with stable components. Z. Wahrsch. Verw. Gebiete 12, 267–289 (1969)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Rogers, C., Taylor, S.J.: Functions continuous and singular with respect to a Hausdorff measure. Mathematika 8, 1–31 (1961)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Talagrand, M.: Hausdorff measure of trajectories of multiparameter fractional Brownian motion. Ann. Probab. 23, 767–775 (1995)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Taylor, S.J.: The measure theory of random fractals. Math. Proc. Camb. Philos. Soc 100, 383–406 (1986)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Xiao, Y.: Hausdorff measure of sample paths of Gaussian random fields. Osaka J. Math. 33, 895–913 (1996)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Xiao, Y.: Hausdorff measure of the graph of fractional Brownian motion. Math. Proc. Camb. Philo. Soc. 122, 565–576 (1997)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Xiao, Y.: Random fractals and Markov processes. In: Lapidus, M.L., van Frankenhuijsen, M. (eds.) Fractal Geometry and Application: A Jubilee of Benoit Mandelbrot, pp. 261–338. American Mathematical Society, Providence (2004)CrossRefGoogle Scholar
  19. 19.
    Xiao, Y.: Sample path properties of anisotropic Gaussian random fields. In: Khoshnevisan, D., Rassoul-Agha, F. (eds.) A Minicourse on Stochastic Partial Differential Equations, Lecture Notes in Mathmatics, vol. 1962, pp. 145–212. Springer, New York (2009)CrossRefGoogle Scholar
  20. 20.
    Xiao, Y.: Recent developments on fractal properties of Gaussian random fields. In: Barral, J., Seuret, S. (eds.) Further Developments in Fractals and Related Fields, pp. 255–288. Springer, New York (2013)CrossRefGoogle Scholar
  21. 21.
    Xue, Y., Xiao, Y.: Fractal and smoothness properties of anisotropic Gaussian models. Front. Math. China 6, 1217–1246 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Yaglom, A.M.: Some classes of random fields in \(n\)-dimensional space, related to stationary processes. Theory Probab. Appl. 2, 273–320 (1957)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Statistics and MathematicsZhejiang Gongshang UniversityHangzhouChina
  2. 2.School of ScienceJimei UniversityXiamenChina

Personalised recommendations