Probability Theory and Related Fields

, Volume 127, Issue 2, pp 186–204 | Cite as

Stochastic evolution equations with fractional Brownian motion

  • S. Tindel
  • C.A. Tudor
  • F. Viens


In this paper linear stochastic evolution equations driven by infinite-dimensional fractional Brownian motion are studied. A necessary and sufficient condition for the existence and uniqueness of the solution is established and the spatial regularity of the solution is analyzed; separate proofs are required for the cases of Hurst parameter above and below 1/2. The particular case of the Laplacian on the circle is discussed in detail.

Key words or phrases:

Fractional Brownian motion Stochastic partial differential equation Hurst parameter 


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  1. 1.
    Alos, E., Mazet, O., Nualart, D.: Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29, 766–801 (1999)zbMATHGoogle Scholar
  2. 2.
    Coutin, L., Decreusefond, L.: Stochastic differential equations driven by a fractional Brownian motion. Preprint, 1999Google Scholar
  3. 3.
    DaPrato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. Cambridge Univ. Press, 1992Google Scholar
  4. 4.
    Decreusefond, L., Ustunel, A.-S.: Stochastic analysis of the fractional Brownian motion. Potential Anal. 10, 177–214 (1997)CrossRefzbMATHGoogle Scholar
  5. 5.
    Grecksch, W., Ahn, V.V.: A parabolic stochastic differential equation with fractional Brownian motion input. Stat. Probab. Lett. 41, 337–346 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Duncan, T.E., Maslowski, B., Pasik-Duncan, B.: Fractional Brownian motion and stochastic equations in Hilbert spaces. Preprint, 2001Google Scholar
  7. 7.
    Kleptsyna, M.L., Kloeden, P.E., Ahn, V.V.: Existence and uniqueness theorems for stochastic differential equations with fractal Brownian motion. (Russian) Problemy Peredachi Informatsii 34, 51–61 (1998), translation in Problems Inform. Transmission 34(1998), 332–341 (1999)zbMATHGoogle Scholar
  8. 8.
    Lin, S.J.: Stochastic calculus of fractional Brownian motion. Stochastics and Stochastic Reports 55, 121–140 (1995)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Mandelbrot, B.B., Van Ness, J.W.: Fractional Brownian motion, fractional noises and application. SIAM Rev. 10, 422–437 (1968)zbMATHGoogle Scholar
  10. 10.
    Maslowski, B., Nualart, D.: Stochastic evolution equations driven by fBm. Preprint, 2002Google Scholar
  11. 11.
    Nualart, D.: Malliavin Calculus and Related topics. Springer Verlag, 1995Google Scholar
  12. 12.
    Peszat, S., Zabczyk, J.: Nonlinear stochastic wave and heat equations. Probab. Theory Related Fields 116, 421–443 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Reed, M., Simon, B.: Methods of modern mathematical physics. 2nd ed. Academic Press, Inc., 1980Google Scholar
  14. 14.
    Tindel, S., Viens, F.: On space-time regularity for the stochastic heat equation on Lie groups. J. Func. Analy. 169, 559–603 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Tindel, S., Viens, F.: Regularity conditions for the stochastic heat equation on some Lie groups. Seminar on Stochastic Analysis, Random Fields and Applications III, Centro Stefano Franscini, Ascona, September 1999. Progress in Probability, 52 Birkhäuser, 2002, pp. 275–297Google Scholar
  16. 16.
    Tudor, C.A.: Stochastic calculus with respect to the infinite dimensional fractional Brownian motion. Preprint, 2002Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Département de Mathématiques, Institut GaliléeUniversité de Paris 13VilletaneuseFrance
  2. 2.Laboratoire de ProbabilitésUniversité de Paris 6Paris Cedex 05France
  3. 3.Dept. Mathematics & Dept. StatisticsPurdue UniversityWest LafayetteUSA

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