On the pointwise regularity of the Multifractional Brownian Motion and some extensions
Authors:
C. Esser and L. Loosveldt
Journal:
Theor. Probability and Math. Statist. 110 (2024), 55-73
MSC (2020):
Primary 60G22, 60G17, 26A15, 42C40
DOI:
https://doi.org/10.1090/tpms/1206
Published electronically:
May 10, 2024
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Abstract: We study the pointwise regularity of the Multifractional Brownian Motion and, in particular, we obtain the existence of so-called slow points of the process, that is points which exhibit a slow oscillation instead of the a.e. regularity. This result entails that a non self-similar process can also exhibit such a behavior. We also consider various extensions with the aim of imposing weaker regularity assumptions on the Hurst function without altering the regularity of the process.
References
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References
- A. Ayache, Multifractional stochastic fields: Wavelet strategies in multifractional frameworks, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019. MR 3839281
- A. Ayache and P. R. Bertrand, A process very similar to multifractional Brownian motion, Recent developments in fractals and related fields, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2010, pp. 311–326. MR 2743002
- A. Ayache and Y. Esmili, Wavelet-type expansion of the generalized Rosenblatt process and its rate of convergence, J. Fourier Anal. Appl. 26 (2020), no. 3, Paper No. 51, 35. MR 4110623
- A. Ayache, C. Esser, and T. Kleyntssens, Different possible behaviors of wavelet leaders of the Brownian motion, Statist. Probab. Lett. 150 (2019), 54–60. MR 3922488
- A. Ayache and M. S. Taqqu, Rate optimality of wavelet series approximations of fractional Brownian motion, J. Fourier Anal. Appl. 9 (2003), no. 5, 451–471. MR 2027888
- —, Multifractional processes with random exponent, Publ. Mat. 49 (2005), no. 2, 459–486. MR 2177638
- A. Benassi, P. Bertrand, S. Cohen, and J. Istas, Identification of the Hurst index of a step fractional Brownian motion, vol. 3, 2000, 19th “Rencontres Franco-Belges de Statisticiens” (Marseille, 1998), pp. 101–111. MR 1819289
- A. Benassi, S. Jaffard, and D. Roux, Elliptic Gaussian random processes, Rev. Mat. Iberoamericana 13 (1997), no. 1, 19–90. MR 1462329
- B. Boufoussi, M. Dozzi, and R. Guerbaz, On the local time of multifractional Brownian motion, Stochastics 78 (2006), no. 1, 33–49. MR 2219711
- —, Sample path properties of the local time of multifractional Brownian motion, Bernoulli 13 (2007), no. 3, 849–867. MR 2348754
- A. Cohen, Biorthogonal wavelets, Wavelets, Wavelet Anal. Appl., vol. 2, Academic Press, Boston, MA, 1992, pp. 123–152. MR 1161250
- A. Cohen, I. Daubechies, and J.-C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. 45 (1992), no. 5, 485–560. MR 1162365
- S. Cohen, From self-similarity to local self-similarity: the estimation problem, Fractals: theory and applications in engineering, Springer, London, 1999, pp. 3–16. MR 1726364
- K. Daoudi, J. Lévy Véhel, and Y. Meyer, Construction of continuous functions with prescribed local regularity, Constr. Approx. 14 (1998), no. 3, 349–385. MR 1626706
- I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, 1992. MR 1162107
- L. Daw and L. Loosveldt, Wavelet methods to study the pointwise regularity of the generalized Rosenblatt process, Electron. J. Probab. 27 (2022), 1–45. MR 4515708
- G. Didier, S. Jaffard, and V. Pipiras, On the vaguelet and Riesz properties of $L^2$-unbounded transformations of orthogonal wavelet bases, J. Approx. Theory 176 (2013), 94–117. MR 3119252
- D. L. Donoho, Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition, Appl. Comput. Harmon. Anal. 2 (1995), no. 2, 101–126. MR 1325535
- C. Esser and L. Loosveldt, Slow, ordinary and rapid points for Gaussian Wavelets Series and application to Fractional Brownian Motions, ALEA Lat. Am. J. Probab. Math. Stat. 19 (2022), no. 2, 1471–1495. MR 4517730
- K. J. Falconer, Tangent fields and the local structure of random fields, J. Theoret. Probab. 15 (2002), no. 3, 731–750. MR 1922445
- —, The local structure of random processes, J. London Math. Soc. (2) 67 (2003), no. 3, 657–672. MR 1967698
- J.-P. Kahane, Some random series of functions, second ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. MR 833073
- T. Kleyntssens and S. Nicolay, From the Brownian motion to a multifractal process using the Lévy–Ciesielski construction, Statist. Probab. Lett. 186 (2022), Paper No. 109450, 5. MR 4398453
- P. G. Lemarié and Y. Meyer, Ondelettes et bases hilbertiennes, Rev. Mat. Iberoamericana 2 (1986), no. 1-2, 1–18. MR 864650
- Y. Meyer and D. Salinger, Wavelets and Operators, vol. 1, Cambridge University Press, 1995. MR 1228209
- R. F. Peltier and J. Lévy Véhel, Multifractional Brownian motion: Definition and preliminary results, Rapport de recherche de l’INRIA 2645 (1995).
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Additional Information
C. Esser
Affiliation:
Université de Liège, Département de mathématique – zone Polytech 1, 12 allée de la Découverte, Bât. B37, B-4000 Liège
Email:
celine.esser@uliege.be
L. Loosveldt
Affiliation:
Université de Liège, Département de mathématique – zone Polytech 1, 12 allée de la Découverte, Bât. B37, B-4000 Liège
Email:
l.loosveldt@uliege.be
Keywords:
Multifractional Brownian motion,
random wavelets series,
modulus of continuity,
slow/ordinary/rapid points
Received by editor(s):
February 10, 2023
Accepted for publication:
July 27, 2023
Published electronically:
May 10, 2024
Article copyright:
© Copyright 2024
Taras Shevchenko National University of Kyiv
