Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Antoine Ayache, Narn-Rueih Shieh, and Yimin Xiao

Full-text: Open access

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.

Résumé

Au moyen d’une méthode d’ondelettes nous montrons que le mouvement Brownien multifractionnaire de type harmonisable à N indices (mfBm) est un champ Gaussien localement non-déterministe. Grâce à cette propriété nous établissons ensuite la bicontinuité des temps locaux d’un (N, d)-mfBm et cela nous permet d’obtenir de nouveaux résultats concernant son comportement trajectoriel.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 4 (2011), 1029-1054.

Dates
First available in Project Euclid: 6 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1317906500

Digital Object Identifier
doi:10.1214/10-AIHP408

Mathematical Reviews number (MathSciNet)
MR2884223

Zentralblatt MATH identifier
1268.60048

Subjects
Primary: 60G15: Gaussian processes 60G17: Sample path properties 28A80: Fractals [See also 37Fxx]

Keywords
Multifractional Brownian motion Local nondeterminism Local times Joint continuity

Citation

Ayache, Antoine; Shieh, Narn-Rueih; Xiao, Yimin. Multiparameter multifractional Brownian motion: Local nondeterminism and joint continuity of the local times. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 4, 1029--1054. doi:10.1214/10-AIHP408. https://projecteuclid.org/euclid.aihp/1317906500


Export citation

References

  • [1] R. J. Adler. The Geometry of Random Fields. Wiley, New York, 1981.
  • [2] A. Ayache. Hausdorff dimension of the graph of fractional Brownian sheet. Rev. Mat. Iberoamericana 20 (2004) 395–412.
  • [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] A. Ayache, S. Jaffard and M. S. Taqqu. Wavelet construction of Generalized Multifractional Processes. Rev. Mat. Iberoamericana 23 (2007) 327–370.
  • [5] A. Ayache and S. Léger. Fractional and multifractional Brownian sheet. Preprint, 2000.
  • [6] A. Ayache and M. S. Taqqu. Multifractional processes with random exponent. Publ. Mat. 49 (2005) 459–486.
  • [7] A. Ayache and Y. Xiao. Asymptotic growth properties and Hausdorff dimension of fractional Brownian sheets. J. Fourier Anal. Appl. 11 (2005) 407–439.
  • [8] D. Baraka, T. Mountford and Y. Xiao. Hölder properties of local times for fractional Brownian motions. Metrika 69 (2009) 125–152.
  • [9] A. Benassi, S. Jaffard and D. Roux. Elliptic Gaussian random processes. Rev. Mat. Iberoamericana 13 (1997) 19–90.
  • [10] S. M. Berman. Gaussian sample function: Uniform dimension and Hölder conditions nowhere. Nagoya Math. J. 46 (1972) 63–86.
  • [11] S. M. Berman. Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J. 23 (1973) 69–94.
  • [12] B. Boufoussi, M. Dozzi and R. Guerbaz. On the local time of multifractional Brownian motion. Stochastics 78 (2006) 33–49.
  • [13] B. Boufoussi, M. Dozzi and R. Guerbaz. Sample path properties of the local time of multifractional Brownian motion. Bernoulli 13 (2007) 849–867.
  • [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.
  • [15] J. Cuzick. Local nondeterminism and the zeros of Gaussian processes. Ann. Probab. 6 (1978) 72–84.
  • [16] I. Daubechies. Ten Lectures on Wavelets. CBMS-NSF Regional Conf. Ser. in Appl. Math. 61. SIAM, Philadelphia, 1992.
  • [17] W. Ehm. Sample function properties of multi-parameter stable processes. Z. Wahrsch. Verw. Gebiete 56 (1981) 195–228.
  • [18] D. Geman and J. Horowitz. Occupation densities. Ann. Probab. 8 (1980) 1–67.
  • [19] E. Herbin. From N parameter fractional Brownian motions to N parameter multifractional Brownian motions. Rocky Mountain J. Math. 36 (2006) 1249–1284.
  • [20] J.-P. Kahane. Some Random Series of Functions, 2nd edition. Cambridge Univ. Press, Cambridge, 1985.
  • [21] D. Khoshnevisan. Multiparameter Processes: An Introduction to Random Fields. Springer, New York, 2002.
  • [22] P. G. Lemarié and Y. Meyer. Ondelettes et bases hilbertiennes. Rev. Mat. Iberoamericana 2 (1986) 1–18.
  • [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] N. Luan and Y. Xiao. Spectral conditions for strong local nondeterminism and exact Hausdorff measure of ranges of Gaussian random fields. Preprint, 2010.
  • [25] M. M. Meerschaert, D. Wu and Y. Xiao. Local times of multifractional Brownian sheets. Bernoulli 13 (2008) 865–898.
  • [26] Y. Meyer. Wavelets and Operators, Vol. 1. Cambridge Univ. Press, Cambridge, 1992.
  • [27] D. Monrad and L. D. Pitt. Local nondeterminism and Hausdorff dimension. In Seminar on Stochastic Processes 1986 163–189. E. Cinlar, K. L. Chung and R. K. Getoor (Eds). Prog. Probab. Statist. Birkhäuser, Boston, MA, 1987.
  • [28] L. D. Pitt. Local times for Gaussian vector fields. Indiana Univ. Math. J. 27 (1978) 309–330.
  • [29] S. A. Stoev and M. S. Taqqu. How rich is the class of multifractional Brownian motions? Stochastic Process. Appl. 116 (2006) 200–221.
  • [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.
  • [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.