Potential Analysis

, Volume 36, Issue 1, pp 1–34 | Cite as

The Stochastic Wave Equation with Multiplicative Fractional Noise: A Malliavin Calculus Approach

  • Raluca M. Balan


We consider the stochastic wave equation with multiplicative noise, which is fractional in time with index H > 1/2, and has a homogeneous spatial covariance structure given by the Riesz kernel of order α. The solution is interpreted using the Skorohod integral. We show that the sufficient condition for the existence of the solution is α > d − 2, which coincides with the condition obtained in Dalang (Electr J Probab 4(6):29, 1999), when the noise is white in time. Under this condition, we obtain estimates for the p-th moments of the solution, we deduce its Hölder continuity, and we show that the solution is Malliavin differentiable of any order. When d ≤ 2, we prove that the first-order Malliavin derivative of the solution satisfies a certain integral equation.


Stochastic wave equation Fractional Brownian motion Spatially homogeneous Gaussian noise Malliavin calculus 

Mathematics Subject Classifications (2010)

Primary 60H15; Secondary 60H07 


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  1. 1.
    Balan, R.M., Tudor, C.A.: Stochastic heat equation with multiplicative fractional-colored noise. J. Theor. Probab. 23, 834–870 (2010)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Balan, R.M., Tudor, C.A.: The stochastic wave equation with fractional noise: a random field approach. Stoch. Proc. Their Appl. 120, 2468–2494 (2010)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Conus, D., Dalang, R.C.: The non-linear stochastic wave equation in high dimensions. Electr. J. Probab. 22, 629–670 (2009)MathSciNetGoogle Scholar
  4. 4.
    Dalang, R.C.: Extending martingale measure stochastic integral with applications to spatially homogenous s.p.d.e.’s. Electr. J. Probab. 4(6), 29 (1999) (Erratum in Electr. J. Probab. 6, 5 (2001))MathSciNetGoogle Scholar
  5. 5.
    Hu, Y.: Heat equations with fractional white noise potentials. Appl. Math. Optim. 43, 221–243 (2001)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Hu, Y., Nualart, D.: Stochastic heat equation driven by fractional noise and local time. Probab. Theory Relat. Fields 143, 285–328 (2009)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Hu, Y., Nualart, D., Song, J.: Feynman–Kac formula for heat equation driven by fractional white noise. Ann. Probab. 39, 291–326 (2011)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1991)CrossRefMATHGoogle Scholar
  9. 9.
    Millet, A., Sanz-Solé, S.: The stochastic wave equation in two space dimension: smoothness of the law. Ann. Probab. 27, 803–844 (1999)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Nualart, D.: Malliavin Calculus and Related Topics, 2nd edn. Springer, Berlin (2006)MATHGoogle Scholar
  11. 11.
    Quer-Sardanyons, L., Tindel, S.: The 1-d stochastic wave equation driven by a fractional Brownian motion. Stoch. Process. Their Appl. 117, 1448–1472 (2007)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Sanz-Solé, M.: Malliavin Calculus with Applications to Stochastic Partial Differential Equations. EPFL Press (2005)Google Scholar
  13. 13.
    Schwartz, L.: Théorie des Distributions. Hermann, Paris (1966)MATHGoogle Scholar
  14. 14.
    Taylor, M.E.: Partial Differential Equations I. Basic Theory. Applied Mathematical Sciences, vol. 115. Springer, New York (1996)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada

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