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From $$N$$ parameter fractional Brownian motions to $$N$$ parameter multifractional Brownian motions. (English) Zbl 1135.60020
Summary: Multifractional Brownian motion is an extension of the well-known fractional Brownian motion where the Hölder regularity is allowed to vary along the paths. In this paper, two kinds of multi-parameter extensions of mBm are studied: one is isotropic while the other is not. For each of these processes, a moving average representation, a harmonizable representation, and the covariance structure are given.
The Hölder regularity is then studied. In particular, the case of an irregular exponent function $$H$$ is investigated. In this situation, the almost sure pointwise and local Holder exponents of the multi-parameter mBm are proved to be equal to the correspondent exponents of $$H$$. Eventually, a local asymptotic self-similarity property is proved. The limit process can be another process than fBm.

##### MSC:
 60G15 Gaussian processes 60G05 Foundations of stochastic processes
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##### References:
 [1] A. Ayache, S. Cohen and J. Lévy Véhel, The covariance structure of multifractional Brownian motion, with application to long range dependence , ICASSP, 2000. [2] A. Ayache and J. Lévy Véhel, Generalized multifractional Brownian motion : Definition and preliminary results , in Fractals : Theory and application in engineering (M. Dekking, J. Lévy-Véhel, E. Lutton and C. Tricot, eds.), Springer, New York, 1999. · Zbl 0964.60046 [3] ——–, Generalized multifractional Brownian motion , SISP 3 (2000), 7-18. · Zbl 0979.60023 · doi:10.1023/A:1009901714819 [4] A. Benassi, S. Jaffard and D. Roux, Elliptic Gaussian random processes , Rev. Mat. Iberoamericana 13 (1998), 19-89. · Zbl 0880.60053 · doi:10.4171/RMI/217 · eudml:39530 [5] P. Billingsley, Convergence of probability measures , in Wiley series in probability and statistics , 2nd ed., Wiley, New York, 1999. · Zbl 0944.60003 [6] S. Cohen, From self-similarity to local self-similarity : The estimation problem , in Fractals : Theory and application in engineering (M. Dekking, J. Lévy-Véhel, E. Lutton and C. Tricot, eds.), Springer, New York, 1999. · Zbl 0965.60073 [7] E. Herbin and J. Lévy Véhel, Fine analysis of the regularity of Gaussian processes : Stochastic $$2$$-microlocal analysis , · Zbl 1175.60032 · doi:10.1016/j.spa.2008.11.005 [8] O. Kallenberg, Foundations of modern probability , Springer, New York, 1997. · Zbl 0892.60001 [9] I. Karatzas and S. Shreve, Brownian motion and stochastic calculus , Springer, New York, 1991. · Zbl 0734.60060 [10] D. Khoshnevisan, Multiparameter processes, An introduction to random fields , Springer, New York, 2002. · Zbl 1005.60005 · doi:10.1007/b97363 [11] S. Léger and M. Pontier, Drap brownien fractionnaire , Note aux CRAS, Paris, série I, mathématiques 329 (1999), 893-898. · Zbl 0945.60047 · doi:10.1016/S0764-4442(00)87495-9 [12] T. Lindstrom, Fractional Brownian fields as integrals of white noise , Bull. London Math. Soc. 25 (1993), 83-88. · Zbl 0741.60031 · doi:10.1112/blms/25.1.83 [13] R. Peltier and J. Lévy-Véhel, Multifractional Brownian motion : Definition and preliminary results , Rapport de recherche INRIA 2645, 1995. [14] B. Pesquet-Popescu, Modélisation bidimensionnelle de processus non stationnaires et application à l’étude du fond sous-marin , thèse de l’ENS Cachan, 1998.
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