Gaussian Fields Satisfying Simultaneous Operator Scaling Relations

  • Marianne Clausel
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


In this chapter we define a special class of group of self-similar Gaussian fields. We present a harmonizable representation of m-parameter group self-similar Gaussian fields by utilizing the Haar measure of this group. These fields also have stationary rectangular increments according to special directions linked to coreduction of matrices of the considered m-parameter group.


Haar Measure Fractional Brownian Motion Positive Real Part Gaussian Field Hurst Index 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ayache, A., Léger, S., Pontier, M.: Drap Brownien fractionnaire, Pot. Anal. 17 31–43 (2002)MATHCrossRefGoogle Scholar
  2. 2.
    Benson, D., Meerschaert, M.M., Bäumer, B., Scheffler, H.P.: Aquifer operator-scaling and the effect on solute mixing and dispersion, Water Resour. Res. 42 W01415, 1–18 (2006)Google Scholar
  3. 3.
    Biermé, H., Meerschaert, M.M., Scheffler, H.P.: Operator scaling stable random fields, Stoch. Proc. App. 117 312–332 (2007)MATHCrossRefGoogle Scholar
  4. 4.
    Bonami, A., Estrade, A.: Anisotropic analysis of some Gaussian models, J. F. Anal. Appl. 9 215–236 (2003)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Clausel, M.: Etude de quelques notions d’irrégularité: le point de vue ondelettes. MA Thesis, Université Paris XII (2008)Google Scholar
  6. 6.
    Davies, S., Hall, P.: Fractal analysis of surface roughness by using spatial data (with discussion). J. Roy. Statist. Soc. Ser. B 61 3–37 (1999)Google Scholar
  7. 7.
    Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces, Academic, San Diego, CA (1978)MATHGoogle Scholar
  8. 8.
    Hudson, W.N., Mason, J.D.: Operator-self-similar processes in a finite dimensional space, Trans. Am. Math. Soc. 34 3–35 (1982)MathSciNetGoogle Scholar
  9. 9.
    Kamont, A.: On the fractional anisotropic Wiener field, Prob. Math. Stat. 16(1) 85–98 (1996)MATHMathSciNetGoogle Scholar
  10. 10.
    Kolmogorov, A.N.: Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum, C.R.A.S. de l’URSS 26 115–118 (1940)Google Scholar
  11. 11.
    Kolodynski, S., Rosinski, J.: Group self-similar stable processes in d, J. Theor. Prob. 16(4) 855–876 (2003)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lamperti, J.W.: Semi-stable stochastic processes, Trans. Am. Math. Soc. 104 62–78 (1962)MATHMathSciNetGoogle Scholar
  13. 13.
    Lemarié-Rieusset, P.G.: Projecteurs invariants, matrices de dilatation, ondelettes et analyses multi-résolutions, Rev. Mat. Iber. 10 283–347 (1994)MATHGoogle Scholar
  14. 14.
    Mandelbrot, B., Van Ness, J.: Fractional Brownian motion, fractional noises and applications, Siam Rev. 10 422–437 (1968)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Meerschaert, M., Scheffler, H.P.: Limit Distributions for Sums of Independent Random Vectors, Wiley Series in Probability and Statistics, Wiley, New York (2001)MATHGoogle Scholar
  16. 16.
    Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes, Chapman & Hall, New York (1994)MATHGoogle Scholar
  17. 17.
    Schertzer, D., Lovejoy, S.: Generalised scale invariance in turbulent phenomena, Phys. Chem. Hydrodyn. J. 6 623–635 (1985)Google Scholar
  18. 18.
    Schertzer, D., Lovejoy, S.: Physically based rain and cloud modeling by anisotropic, multiplicative turbulent cascades, J. Geophys. Res. 92(D8) 9693–9714 (1987)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Université Paris XIICréteilFrance

Personalised recommendations