[en] We study the Hölderian regularity of Gaussian wavelets series and show that
they display, almost surely, three types of points: slow, ordinary and rapid.
In particular, this fact holds for the Fractional Brownian Motion. We also show
that this property is satisfied for a multifractal extension of Gaussian
wavelet series. Finally, we remark that the existence of slow points is
specific to these functions.
Disciplines :
Mathematics
Author, co-author :
Esser, Céline ; Université de Liège - ULiège > Département de mathématique > Analyse mathématique et ses interactions avec la théorie des probabilités
Loosveldt, Laurent ; Université de Liège - ULiège > Département de mathématique > Analyse - Analyse fonctionnelle - Ondelettes
Language :
English
Title :
Slow, ordinary and rapid points for Gaussian Wavelets Series and application to Fractional Brownian Motions
Publication date :
2022
Journal title :
ALEA: Latin American Journal of Probability and Mathematical Statistics
eISSN :
1980-0436
Publisher :
Instituto Nacional de Matematica Pura e Aplicada, Brazil
Aubry, J.-M. and Bastin, F. A walk from multifractal analysis to functional analysis with Sv spaces, and back. In Recent developments in fractals and related fields, Appl. Numer. Harmon. Anal., pp. 93-106. Birkhauser Boston, Boston, MA (2010). MR2742989.
Ayache, A. Multifractional stochastic fields. Wavelet strategies in multifractional frameworks. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2019). ISBN 978-981-4525-65-7. MR3839281.
Ayache, A. and Bertrand, P. R. A process very similar to multifractional Brownian motion. In Recent developments in fractals and related fields, Appl. Numer. Harmon. Anal., pp. 311-326. Birkhauser Boston, Boston, MA (2010). MR2743002.
Ayache, A., Esser, C., and Kleyntssens, T. Different possible behaviors of wavelet leaders of the Brownian motion. Statist. Probab. Lett., 150, 54-60 (2019). MR3922488.
Ayache, A. and Taqqu, M. S. Rate optimality of wavelet series approximations of fractional Brownian motion. J. Fourier Anal. Appl., 9 (5), 451-471 (2003). MR2027888.
Bastin, F., Esser, C., and Jaffard, S. Large deviation spectra based on wavelet leaders. Rev. Mat. Iberoam., 32 (3), 859-890 (2016). MR3556054.
Bingham, N. H. Variants on the law of the iterated logarithm. Bull. London Math. Soc., 18 (5), 433-467 (1986). MR847984.
Christensen, J. P. R. Topology and Borel structure. Descriptive topology and set theory with applications to functional analysis and measure theory. North-Holland Mathematics Studies, Vol. 10. North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York (1974). MR0348724.
Clausel, M. and Nicolay, S. Some prevalent results about strongly monoHolder functions. Nonlinearity, 23 (9), 2101-2116 (2010). MR2672638.
Clausel, M. and Nicolay, S. Wavelets techniques for pointwise anti-Holderian irregularity. Constr. Approx., 33 (1), 41-75 (2011). MR2747056.
Cohen, A., Daubechies, I., and Feauveau, J.-C. Biorthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math., 45 (5), 485-560 (1992). MR1162365.
Daubechies, I. Ten lectures on wavelets, volume 61 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1992). ISBN 0-89871-274-2. MR1162107.
Esser, C. and Jaffard, S. Divergence of wavelet series: a multifractal analysis. Adv. Math., 328, 928-958 (2018). MR3771146.
Esser, C., Kleyntssens, T., and Nicolay, S. A multifractal formalism for non-concave and nonincreasing spectra: the leaders profile method. Appl. Comput. Harmon. Anal., 43 (2), 269-291 (2017). MR3668039.
Hunt, B. R., Sauer, T., and Yorke, J. A. Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces. Bull. Amer. Math. Soc. (N.S.), 27 (2), 217-238 (1992). MR1161274.
Jaffard, S. Beyond Besov spaces. I. Distributions of wavelet coefficients. J. Fourier Anal. Appl., 10 (3), 221-246 (2004a). MR2066421.
Jaffard, S. Wavelet techniques in multifractal analysis. In Fractal geometry and applications: a jubilee of Benoît Mandelbrot, Part 2, volume 72 of Proc. Sympos. Pure Math., pp. 91-151. Amer. Math. Soc., Providence, RI (2004b). MR2112122.
Jaffard, S. and Meyer, Y. Wavelet methods for pointwise regularity and local oscillations of functions. Mem. Amer. Math. Soc., 123 (587), x+110 (1996). MR1342019.
Kahane, J.-P. Some random series of functions, volume 5 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition (1985). ISBN 0-521-24966-X; 0-521-45602-9. MR833073.
Khintchine, A. Über einen Satz der Wahrscheinlichkeitsrechnung. Fund. Math., 6, 9-20 (1924). DOI: 10.4064/fm-6-1-9-20.
Khoshnevisan, D. and Shi, Z. Fast sets and points for fractional Brownian motion. In Séminaire de Probabilités XXXIV, pp. 393-416. Springer Berlin Heidelberg (2000). DOI: 10.1007/BFb0103816.
Kreit, D. and Nicolay, S. Generalized pointwise Holder spaces defined via admissible sequences. J. Funct. Spaces, pp. Art. ID 8276258, 11 (2018). MR3812831.
Loosveldt, L. and Nicolay, S. Generalized Tf spaces: on the trail of Calderón and Zygmund. Dissertationes Math., 554, 64 (2020). MR4166514.
Loosveldt, L. and Nicolay, S. Generalized spaces of pointwise regularity: toward a general framework forthe WLM. Nonlinearity, 34 (9), 6561-6586 (2021). MR4304490.
Mallat, S. A Wavelet Tour of Signal Processing. Academic Press (2009). ISBN 978-0-12-374370-1. DOI: 10.1016/B978-0-12-374370-1.X0001-8.
Marcus, M. B. Holder conditions for Gaussian processes with stationary increments. Trans. Amer. Math. Soc., 134, 29-52 (1968). MR230368.
Marcus, M. B. and Rosen, J. Moduli of continuity of local times of strongly symmetric Markov processes via Gaussian processes. J. Theoret. Probab., 5 (4), 791-825 (1992). MR1182681.
Meyer, Y. Wavelets and operators, volume 37 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1992). ISBN 0-521-42000-8; 0-521-45869-2. Translated from the 1990 French original by D. H. Salinger. MR1228209.
Meyer, Y., Sellan, F., and Taqqu, M. S. Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motion. J. Fourier Anal. Appl., 5 (5), 465-494 (1999). MR1755100.
Monrad, D. and Rootzén, H. Small values of Gaussian processes and functional laws of the iterated logarithm. Probab. Theory Related Fields, 101 (2), 173-192 (1995). MR1318191.
Orey, S. Growth rate of certain Gaussian processes. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory, pp. 443-451. Univ. California Press, Berkeley, Calif. (1972). MR0402897.
Orey, S. and Taylor, S. J. How often on a Brownian path does the law of iterated logarithm fail? Proc. London Math. Soc. (3), 28, 174-192 (1974). MR359031.
