Multiparameter multifractional brownian motion : local nondeterminism and joint continuity of the local times

Antoine Ayache; Narn-Rueih Shieh; Yimin Xiao

Annales de l'I.H.P. Probabilités et statistiques (2011)

  • Volume: 47, Issue: 4, page 1029-1054
  • ISSN: 0246-0203

Abstract

top
By using a wavelet method we prove that the harmonisable-type N-parameter multifractional brownian motion (mfBm) is a locally nondeterministic gaussian random field. This nice property then allows us to establish joint continuity of the local times of an (N, d)-mfBm and to obtain some new results concerning its sample path behavior.

How to cite

top

Ayache, Antoine, Shieh, Narn-Rueih, and Xiao, Yimin. "Multiparameter multifractional brownian motion : local nondeterminism and joint continuity of the local times." Annales de l'I.H.P. Probabilités et statistiques 47.4 (2011): 1029-1054. <http://eudml.org/doc/243861>.

@article{Ayache2011,
abstract = {By using a wavelet method we prove that the harmonisable-type N-parameter multifractional brownian motion (mfBm) is a locally nondeterministic gaussian random field. This nice property then allows us to establish joint continuity of the local times of an (N, d)-mfBm and to obtain some new results concerning its sample path behavior.},
author = {Ayache, Antoine, Shieh, Narn-Rueih, Xiao, Yimin},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {multifractional brownian motion; local nondeterminism; local times; joint continuity; multifractional Brownian motion},
language = {eng},
number = {4},
pages = {1029-1054},
publisher = {Gauthier-Villars},
title = {Multiparameter multifractional brownian motion : local nondeterminism and joint continuity of the local times},
url = {http://eudml.org/doc/243861},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Ayache, Antoine
AU - Shieh, Narn-Rueih
AU - Xiao, Yimin
TI - Multiparameter multifractional brownian motion : local nondeterminism and joint continuity of the local times
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2011
PB - Gauthier-Villars
VL - 47
IS - 4
SP - 1029
EP - 1054
AB - By using a wavelet method we prove that the harmonisable-type N-parameter multifractional brownian motion (mfBm) is a locally nondeterministic gaussian random field. This nice property then allows us to establish joint continuity of the local times of an (N, d)-mfBm and to obtain some new results concerning its sample path behavior.
LA - eng
KW - multifractional brownian motion; local nondeterminism; local times; joint continuity; multifractional Brownian motion
UR - http://eudml.org/doc/243861
ER -

References

top
  1. [1] R. J. Adler. The Geometry of Random Fields. Wiley, New York, 1981. Zbl0478.60059MR611857
  2. [2] A. Ayache. Hausdorff dimension of the graph of fractional Brownian sheet. Rev. Mat. Iberoamericana 20 (2004) 395–412. Zbl1057.60033MR2073125
  3. [3] A. Ayache, S. Cohen and J. Lévy-Véhel. The covariance structure of multifractional Brownian motion, with application to long range dependence. In Proceeding of ICASSP, Istambul, 2002. 
  4. [4] A. Ayache, S. Jaffard and M. S. Taqqu. Wavelet construction of Generalized Multifractional Processes. Rev. Mat. Iberoamericana 23 (2007) 327–370. Zbl1123.60022MR2351137
  5. [5] A. Ayache and S. Léger. Fractional and multifractional Brownian sheet. Preprint, 2000. Zbl1006.60029
  6. [6] A. Ayache and M. S. Taqqu. Multifractional processes with random exponent. Publ. Mat. 49 (2005) 459–486. Zbl1082.60032MR2177638
  7. [7] A. Ayache and Y. Xiao. Asymptotic growth properties and Hausdorff dimension of fractional Brownian sheets. J. Fourier Anal. Appl. 11 (2005) 407–439. Zbl1088.60033MR2169474
  8. [8] D. Baraka, T. Mountford and Y. Xiao. Hölder properties of local times for fractional Brownian motions. Metrika 69 (2009) 125–152. Zbl06493840MR2481918
  9. [9] A. Benassi, S. Jaffard and D. Roux. Elliptic Gaussian random processes. Rev. Mat. Iberoamericana 13 (1997) 19–90. Zbl0880.60053MR1462329
  10. [10] S. M. Berman. Gaussian sample function: Uniform dimension and Hölder conditions nowhere. Nagoya Math. J. 46 (1972) 63–86. Zbl0246.60038MR307320
  11. [11] S. M. Berman. Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J. 23 (1973) 69–94. Zbl0264.60024MR317397
  12. [12] B. Boufoussi, M. Dozzi and R. Guerbaz. On the local time of multifractional Brownian motion. Stochastics 78 (2006) 33–49. Zbl1124.60061MR2219711
  13. [13] B. Boufoussi, M. Dozzi and R. Guerbaz. Sample path properties of the local time of multifractional Brownian motion. Bernoulli 13 (2007) 849–867. Zbl1138.60032MR2348754
  14. [14] B. Boufoussi, M. Dozzi and R. Guerbaz. Path properties of a class of locally asymptotically self similar processes. Electron. J. Probab. 13 (2008) 898–921. Zbl1191.60046MR2413288
  15. [15] J. Cuzick. Local nondeterminism and the zeros of Gaussian processes. Ann. Probab. 6 (1978) 72–84. Zbl0374.60051MR488252
  16. [16] I. Daubechies. Ten Lectures on Wavelets. CBMS-NSF Regional Conf. Ser. in Appl. Math. 61. SIAM, Philadelphia, 1992. Zbl0776.42018MR1162107
  17. [17] W. Ehm. Sample function properties of multi-parameter stable processes. Z. Wahrsch. Verw. Gebiete 56 (1981) 195–228. Zbl0471.60046MR618272
  18. [18] D. Geman and J. Horowitz. Occupation densities. Ann. Probab. 8 (1980) 1–67. Zbl0499.60081MR556414
  19. [19] E. Herbin. From N parameter fractional Brownian motions to N parameter multifractional Brownian motions. Rocky Mountain J. Math. 36 (2006) 1249–1284. Zbl1135.60020MR2274895
  20. [20] J.-P. Kahane. Some Random Series of Functions, 2nd edition. Cambridge Univ. Press, Cambridge, 1985. Zbl0805.60007MR833073
  21. [21] D. Khoshnevisan. Multiparameter Processes: An Introduction to Random Fields. Springer, New York, 2002. Zbl1005.60005MR1914748
  22. [22] P. G. Lemarié and Y. Meyer. Ondelettes et bases hilbertiennes. Rev. Mat. Iberoamericana 2 (1986) 1–18. Zbl0657.42028MR864650
  23. [23] J. Lévy-Véhel and R. F. Peltier. Multifractional Brownian motion: Definition and preliminary results. Technical Report RR-2645, Institut National de Recherche en Informatique et Automatique, INRIA, Le Chesnay, France, 1995. 
  24. [24] N. Luan and Y. Xiao. Spectral conditions for strong local nondeterminism and exact Hausdorff measure of ranges of Gaussian random fields. Preprint, 2010. Zbl1261.60040MR2885561
  25. [25] M. M. Meerschaert, D. Wu and Y. Xiao. Local times of multifractional Brownian sheets. Bernoulli 13 (2008) 865–898. Zbl1186.60036MR2537815
  26. [26] Y. Meyer. Wavelets and Operators, Vol. 1. Cambridge Univ. Press, Cambridge, 1992. Zbl0776.42019MR1228209
  27. [27] D. Monrad and L. D. Pitt. Local nondeterminism and Hausdorff dimension. In Seminar on Stochastic Processes 1986163–189. E. Cinlar, K. L. Chung and R. K. Getoor (Eds). Prog. Probab. Statist. Birkhäuser, Boston, MA, 1987. Zbl0616.60049MR902433
  28. [28] L. D. Pitt. Local times for Gaussian vector fields. Indiana Univ. Math. J. 27 (1978) 309–330. Zbl0382.60055MR471055
  29. [29] S. A. Stoev and M. S. Taqqu. How rich is the class of multifractional Brownian motions? Stochastic Process. Appl. 116 (2006) 200–221. Zbl1094.60024MR2197974
  30. [30] Y. Xiao. Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields. Probab. Theory Related Fields 109 (1997) 129–157. Zbl0882.60035MR1469923
  31. [31] Y. Xiao. Sample path properties of anisotropic Gaussian random fields. In A Minicourse on Stochastic Partial Differential Equations 145–212. D. Khoshnevisan and F. Rassoul-Agha (Eds). Springer, New York 2009. Zbl1167.60011MR2508776

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.