Skip to main content
SpringerLink
Log in

On Fractional Brownian Motion and Wavelets

  • Published:
Complex Analysis and Operator Theory Aims and scope Submit manuscript

Abstract

Given a fractional Brownian motion (fBm) with Hurst index \({H\in(0,1)}\) , we associate with this a special family of representations of Cuntz algebras related to frequency domains and wavelets. Vice versa, we consider a pair of Haar wavelets satisfying some compatibility conditions, and we construct the covariance functions of fBm with a fixed Hurst index. The Cuntz algebra representations enter the picture as filters of the associated wavelets. Extensions to q-dependent covariance functions leading to a corresponding fBm process will also be discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Partzsch, L.: Vorlesungen zum eindimensionalen Wienerschen Proze  Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 66, p. 112. BSB B. G. Teubner Verlagsgesellschaft, Leipzig (1984)

  2. Nelson E.: Feynman integrals and the Schrödinger equation. J. Math. Phys. 5, 332–343 (1964)

    Article  MATH  Google Scholar 

  3. Crismale V.: Quantum stochastic calculus on interacting Fock spaces: semimartingale estimates and stochastic integral. Commun. Stoch. Anal. 1(2), 321–341 (2007)

    MathSciNet  Google Scholar 

  4. Henkel, M., Schott, R., Stoimenov, S., Unterberger, J.: On the dynamical symmetric algebra of ageing: Lie structure, representations and Appell systems, Quantum probability and infinite dimensional analysis, QP–PQ: Quantum Probab. White Noise Analysis, vol. 20, pp. 233–240. World Scientific Publications, Hackensack (2007)

  5. Unterberger J.: Stochastic calculus for fractional Brownian motion with Hurst exponent \({H > \frac{1}{4}}\) : A rough path method by analytic extension. Ann. Probab. 37, 565–614 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Albeverio, S.: Wiener and Feynman path integrals and their applications. In: Proceedings of N. Wiener Centenary Congress, Proceedings of Symposium on Applied Mathematics, vol. 52, pp. 163–194. AMS, Providence (1997)

  7. Arveson W.: Pure E 0-semigroups and absorbing states. Commun. Math. Phys. 187(1), 19–43 (1987)

    Article  MathSciNet  Google Scholar 

  8. Cho, I., Jorgensen, P.E.T.: C*-algebras generated by partial isometries, J. Appl. Math. Comput. 26, 1–2, 1–48 (2008). ISSN 5865

    Google Scholar 

  9. Choi M.D.: Some assorted inequalities for positive linear maps on C*-algebras. J. Oper. Theory 4(2), 271–285 (1980)

    MATH  Google Scholar 

  10. Bratteli O., Jorgensen P.E.T.: Iterated function systems and permutation representations of the Cuntz algebra. Mem. Am. Math. Soc. 139, 663 (1999)

    MathSciNet  Google Scholar 

  11. Bozejko M., Kummerer B., Speicher R.: q−Gaussian processes: non-commutative and classical aspects. Commun. Math. Phys. 185, 129–154 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jorgensen P.E.T., Schmitt L.M., Werner R.F.: q-canonical commutation relations and stability of the Cuntz algebra. Pacific J. Math. 165, 131–151 (1994)

    MathSciNet  Google Scholar 

  13. Jorgensen P.E.T., Kribs D.W.: Wavelet representations and Fock space on positive matrices. J. Funct. Anal. 197, 526–559 (2003)

    Article  MathSciNet  Google Scholar 

  14. Jorgensen P.E.T., Werner R.F.: Coherent states of the q-canonical commutation relations. Commun. Math. Phys. 164, 455–471 (1994)

    Article  MathSciNet  Google Scholar 

  15. Paolucci, A.M.: Bessel functions and Cuntz algebras representations, APLIMAT 2007, Part II, pp. 109–114

  16. Jorgensen P.E.T., Paolucci A.M.: Wavelets in mathematical physics: q-oscillators. J. Phys. A Math. Gen. 36, 6483–6494 (2003)

    Article  MathSciNet  Google Scholar 

  17. Bratteli O., Jorgensen P.E.T.: Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scale N. Integr. Equ. Oper. Theory 28, 382–443 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Albeverio, S., Jorgensen, P.E.T., Paolucci, A.M.: Multiresolution wavelet analysis of integer scale Bessel functions. J. Math. Phys. 48, 073516 (2007) (24 pages)

    Google Scholar 

  19. Chyzak F., Paule P., Scherzer O., Schoisswohl A., Zimmermann B.: The construction of orthonormal wavelets using symbolic methods and a matrix analytical approach for wavelets on an interval. Exp. Math. 10, 67–86 (2001)

    MathSciNet  MATH  Google Scholar 

  20. Daubechies, I.: Ten Lectures on Wavelets, Vol. 61 in CBMS-NSF. Regional Conference Series in Applied Mathematics (Society for Industrial and Applied Mathematics, Philadelphia) (1992)

  21. Jorgensen, P.E.T.: Analysis and probability: wavelets, signals, fractals. Graduate Texts in Mathematics, vol. 234, pp. xlviii+276. Springer, New York (2006)

  22. Bratteli, O., Jorgensen, P.E.T.: Wavelets through a looking glass. Applied and Numerical Harmonic Analysis, Birkhäuser Boston Inc. Boston, MA, xxii+398 (2002). ISBN 0-8176-4280-3

  23. Bratteli, O., Jorgensen, P.E.T.: Wavelet filters and infinite-dimensional unitary groups. In: Deng, D., Huang, D., Jia, R.-Q., Lin, W., Wang, J. (eds.) Wavelet Analysis and Applications (Guangzhou, China, 1999) AMS/IP Studies in Advanced Mathematics, vol. 25, pp. 35–65. American Mathematical Society, Providence, International Press, Boston (2002)

  24. Bratteli O., Evans D.E., Jorgensen P.E.T.: Compactly supported wavelets and representations of the Cuntz relations. Appl. Comput. Harmon. Anal. 8, 166–196 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jorgensen, P.E.T., Kornelson, K., Shuman, K.: Orthogonal exponentials for Bernoulli iterated function systems. arXiv:math/073385v1 (2007)

  26. Mallat S.G.: Multiresolution approximations and wavelet orthonormal bases of L 2 (R). Trans. Am. Math. Soc. 315, 69–87 (1989)

    MathSciNet  MATH  Google Scholar 

  27. Paule, P., Scherzer, O., Schoisswohl, A.: Wavelets with scale dependent properties, SNCs 2001. LNCS, vol. 2630, pp. 255–265 (2003)

  28. Dutkay, D.E., Jorgensen, P.E.T.: Wavelets on fractals. Rev. Mat. Iberoam. 22(1), 131–180 (2006). ISSN 0213-2230

    Google Scholar 

  29. Jorgensen, P.E.T.: Frame analysis and approximation in reproducing kernel Hilbert spaces. In: Contemprorary Mathematics, vol. 451, pp. 151–169. American Mathematical Society, Providence (2008)

  30. Askey, R., Ismail, M.: Recurrence relations, continued fractions, and orthogonal polynomials. Mem. Amer. Math. Soc. 49, 300 (1984)

    Google Scholar 

  31. Gasper G., Rahman M.: Basic Hypergeometric Series. In: Encyclopedia of Mathematics and Its Applications, Vol. 35. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  32. Ismail M.E.H.: The zeros of basic Bessel functions, the functions J ν+a x (x) and associated orthogonal polynomials. J. Math. Anal. Appl. 86, 1–19 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  33. Jorgensen P.E.T.: Measures in wavelet decompositions. Adv. Appl. Math. 34(3), 561–590 (2005)

    Article  MATH  Google Scholar 

  34. Cooper M.: Dimension, measure and infinite Bernoulli convolutions. Math. Proc. Camb. Phil. Soc. 124, 135–149 (1998)

    Article  MATH  Google Scholar 

  35. Falconer K.: “Fractal geometry” Fractal Geometry Mathematical Foundations and Applications, 2nd edn. Wiley, New York (2003)

    Book  Google Scholar 

  36. Sneddon I.N.: Fourier Transforms. McGraw-Hill, New York (1951)

    Google Scholar 

  37. Sneddon I.N.: Fourier Transforms. Dover, New York (1995)

    Google Scholar 

  38. Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. (Cambridge University Press, Cambridge; The Macmillan Company, New York, 1944; reprinted, Cambridge Mathematical Library, Cambridge University Press, Cambridge (1995)

  39. Ball J.A., Vinnikov V.: Formal reproducing kernel Hilbert spaces: the commutative and noncommutative settings, Reproducing kernel spaces and applications. Oper. Theory Adv. Appl. 143, 77–134 (2003)

    MathSciNet  Google Scholar 

  40. Alpay D., Levanony D.: On the reproducing kernel Hilbert spaces associated with the fractional and bi-fractional Brownian motions. Potential Anal. 28(2), 163–184 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  41. Barnett C., Voliotis A.: A conditional expectation for the full Fock space. J. Oper. Theory 44, 3–23 (2000)

    MathSciNet  MATH  Google Scholar 

  42. Ciesielski Z., Kamont A.: Lévy fractional Brownian random field and function spaces. Acta Sci. Math. 60, 99–118 (1995)

    MathSciNet  MATH  Google Scholar 

  43. Mishura, Y.: Stochastic calculus for fractional Brownian motion and related processes. In: Lecture Notes in Mathematics 1929 (2008)

  44. Dzhaparidze, K.O., Ferreira, J.A.: A frequency domain approach to some results on fractional Brownian motion, CWI report PNA-R0123 (2001)

  45. Dzhaparidze K., van Zanten J.H.: A series expansion of Fractional Brownian Motion. Probab. Theory Relat. Fields 130, 39–55 (2004)

    Article  MATH  Google Scholar 

  46. Nualart D.: Stochastic calculus with respect to the fractional Brownian motion and applications. Contemp. Math. 336, 3–39 (2003)

    Article  MathSciNet  Google Scholar 

  47. Parthasaraty K.R.: An Introduction to Quantum Stochastic Calculus. Birkhauser Verlag, Basel (1992)

    Book  Google Scholar 

  48. Sainty P.: Construction of a complex-valued fractional Brownian motion of order N. J. Math. Phys. 33(9), 3128–3149 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  49. Wiener N.: Differential spaces. J. Math. Phys. Soc. 2, 131–174 (1923)

    Google Scholar 

  50. Paley, R.E.A.C., Wiener, N.: Fourier transforms in a complex domain. AMS, Providence (1934)

  51. Lévy P.: Processus stochastiques et mouvement brownien. Gauthier-Villars, Paris (1948)

    MATH  Google Scholar 

  52. Ciesielski Z.: Hölder condition for realizations of Gaussian processes. Trans. Am. Math. Soc. 99, 403–413 (1961)

    MathSciNet  MATH  Google Scholar 

  53. Itô K., Nisio M.: On the convergence of sums of independent Banach space valued random variables. Osaka J. Math. 5, 35–48 (1968)

    MathSciNet  MATH  Google Scholar 

  54. Huang Z.Y., Li C.J., Wan J.P., Wu Y.: Fractional Brownian motion and sheet as white noise functionals. Acta Math. Sin. (English Series) 22(4), 1183–1188 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  55. Peszat, S., Zabczyk, J.: Stochastic partial differential equations with Lévy Noise: an evolution equation approach. In: Encyclopedia of mathematics and its applications, vol. 113. Cambridge University Press, Cambridge (2007)

  56. Gheondea, A., Kavruk, Ş.: Absolute continuity for operator valued completely positive maps on C*-algebras. J. Math. Phys. 50, 2, (022102,29) (2009)

  57. Cuntz J.: Simple C*-algebras generated by isometries. Commun. Math. Phys. 56, 173–185 (1977)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Bonn, Endenicherallee 60, 53115, Bonn, Germany

    S. Albeverio

  2. HCM, SFB 611, Bonn, BiBoS (Bielefeld-Bonn), IZKS, Bonn, Germany

    S. Albeverio

  3. CERFIM, Locarno, Switzerland

    S. Albeverio

  4. ACC. ARCH (USI), Lugano, Switzerland

    S. Albeverio

  5. Mathematics Department, University of Iowa, Iowa City, IA, 52242, USA

    P. E. T. Jorgensen

  6. Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany

    A. M. Paolucci

Authors
  1. S. Albeverio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. E. T. Jorgensen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. M. Paolucci
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. M. Paolucci.

Additional information

Communicated by Daniel Aron Alpay.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Albeverio, S., Jorgensen, P.E.T. & Paolucci, A.M. On Fractional Brownian Motion and Wavelets. Complex Anal. Oper. Theory 6, 33–63 (2012). https://doi.org/10.1007/s11785-010-0077-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11785-010-0077-2

Keywords

Mathematics Subject Classification (2000)

Search

Navigation