Self-regulating processes

Olivier Barrière (IRCCyN)
Antoine Echelard (Inria and École Centrale Paris)
Jacques Lévy Véhel (Inria and École Centrale Paris)


We construct functions and stochastic processes for which a functional relation holds between amplitude and local regularity, as measured by the pointwise or local Hölder exponent. We consider in particular functions and processes built by extending Weierstrass function, multifractional Brownian motion and the Lévy construction of Brownian motion. Such processes have recently proved to be relevant models in various applications. The aim of this work is to provide a theoretical background to these studies and to provide a first step in the development of a theory for such self-regulating processes.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-30

Publication Date: December 16, 2012

DOI: 10.1214/EJP.v17-2010


  • A. Ayache; J. Lévy Véhel. The generalized multifractional Brownian motion. Stat. Inference Stoch. Process. 3 (2000), no. 1-2, 7--18. MR1819282
  • S. Bianchi; A. Pianese. Modelling stock price movements: multifractality or multifractionality? Quant. Finance 7 (2007), no. 3, 301--319. MR2332737
  • A. Benassi ; S. Jaffard; D. Roux. Elliptic Gaussian random processes. Rev. Mat. Iberoamericana 13 (1997), no. 1, 19--90. MR1462329
  • P. Legrand; J. Lévy Véhel, Holderian regularity-based image interpolation, ICASSP06, International Conference on Acoustics, Speech, and Signal Processing, 2006.
  • K. J. Falconer. The local structure of random processes. J. London Math. Soc. (2) 67 (2003), no. 3, 657--672. MR1967698
  • K.J. Falconer. Localisable, multifractional and multistable processes, Séminaires et Congrès, 28, 1--12, 2012.
  • A.N. Kolmogoroff. A. N. Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum. (German) C. R. (Doklady) Acad. Sci. URSS (N.S.) 26, (1940). 115--118. MR0003441
  • A. Ayache; M. S. Taqqu. Multifractional processes with random exponent. Publ. Mat. 49 (2005), no. 2, 459--486. MR2177638
  • A.Ayache ; S. Cohen; J. Lévy Véhel. The covariance of the multifractional Brownian motion and applications to long-range dependence, ICASSP00, International Conference on Acoustics, Speech, and Signal Processing, 2000.
  • B.B. Mandelbrot; J. .W Van Ness. Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 1968 422--437. MR0242239
  • K. Daoudi; J. Lévy Véhel; Y. Meyer. Construction of continuous functions with prescribed local regularity. Constr. Approx. 14 (1998), no. 3, 349--385. MR1626706
  • R.F. Peltier; J. Lévy Véhel. Multifractional Brownian motion: definition and preliminary results, INRIA Research Report 2645, 1995.
  • S.A. Stoev; M.S. Taqqu. How rich is the class of multifractional Brownian motions? Stochastic Process. Appl. 116 (2006), no. 2, 200--221. MR2197974
  • C. Tricot. Curves and fractal dimension. With a foreword by Michel Mendès France. Translated from the 1993 French original. Springer-Verlag, New York, 1995. xiv+323 pp. ISBN: 0-387-94095-2 MR1302173
  • FracLab, a MatLab toolbox for signal and image analysis. Available at
  • A. Echelard; O. Barrière; J. Lévy Véhel. Terrain modelling with multifractional Brownian motion and self-regulating processes, Lecture Notes in Computer Science, 6374, 342--351, Springer, 2010.
  • O. Barrière; J. Lévy Véhel. Intervalles interbattements cardiaques et processus auto-régulé multifractionnaire. (French) [Application of the self-regulating multifractional process to cardiac interbeat intervals] J. SFdS 150 (2009), no. 1, 54--72. MR2609697
  • A. Echelard; J. Lévy Véhel. Self-regulating processes-based modelling for arrhythmia characterization, ISPHT 2012, International Conference on Imaging and Signal Processing in Health Care and Technology, 2012.
  • A. Echelard; J. Lévy Véhel; A. Philippe. Estimating the self-regulating function of self-regulating midpoint displacement processes, preprint, 2012.
  • R. Fischer; M. Akay; P. Castiglioni; M. Di Rienzo. Multi- and monofractral indices of short-term heart variability, Medical & Biological Engineering & Computing, 5 (41), 543--549, 2003.
  • K. Weierstrass. On Continuous Function of a Real Argument that do not have a Well­-Defined Differential Quotient, Mathematische Werke, 2, 71--74, 1895.
  • G.H. Hardy. Weierstrass's non-differentiable function. Trans. Amer. Math. Soc. 17 (1916), no. 3, 301--325. MR1501044
  • A.T. Wood; G. Chan. Simulation of stationary Gaussian processes in $[0,1]^ d$. J. Comput. Graph. Statist. 3 (1994), no. 4, 409--432. MR1323050

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.