Skip to main content
Log in

Existence and Smoothness of the Density for Spatially Homogeneous SPDEs

  • Published:
Potential Analysis Aims and scope Submit manuscript

Abstract

In this paper, we extend Walsh’s stochastic integral with respect to a Gaussian noise, white in time and with some homogeneous spatial correlation, in order to be able to integrate some random measure-valued processes. This extension turns out to be equivalent to Dalang’s one. Then we study existence and regularity of the density of the probability law for the real-valued mild solution to a general second order stochastic partial differential equation driven by such a noise. For this, we apply the techniques of the Malliavin calculus. Our results apply to the case of the stochastic heat equation in any space dimension and the stochastic wave equation in space dimension d=1,2,3. Moreover, for these particular examples, known results in the literature have been improved.

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

Access this article

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. Bouleau, N., Hirsch, F.: Dirichlet forms and analysis on Wiener space. Walter de Gruyter, Berlin (1991)

    MATH  Google Scholar 

  2. Carmona, R., Nualart, D.: Random nonlinear wave equations: smoothness of the solutions. Probab. Theory Related Fields 79(4), 469–508 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  3. Dalang, R.C.: Extending martingale measure stochastic integral with applications to spatially homogeneous S. P. D. E’s. Electron. J. Probab. 4(6), 29 (1999) (electronic)

    MathSciNet  Google Scholar 

  4. Dalang, R.C., Frangos, N.E.: The stochastic wave equation in two spatial dimensions. Ann. Probab. 26(1), 187–212 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  5. Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. Encyclopedia of mathematics and its applications, vol. 44. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  6. Lévêque, O.: Hyperbolic stochastic partial differential equations driven by boundary noises. Ph.D. thesis, EPFL (2001)

  7. Márquez-Carreras, D., Mellouk, M., Sarrà, M.: On stochastic partial differential equations with spatially correlated noise: smoothness of the law. Stochastic Process. Appl. 93, 269–284 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Millet, A., Sanz-Solé, M.: A stochastic wave equation in two space dimensions: smoothness of the law. Ann. Probab. 27(2), 803–844 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Nualart, D.: The Malliavin calculus and related topics, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  10. Peszat, S.: The Cauchy problem for a nonlinear stochastic wave equation in any dimension. J. Evol. Equ. 2(3), 383–394 (2002)

    Article  MathSciNet  Google Scholar 

  11. Peszat, S., Zabczyk, J.: Nonlinear stochastic wave and heat equations. Probab. Theory Related Fields 116(3), 421–443 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Quer-Sardanyons, L.: The stochastic wave equation: study of the law and approximations. Ph.D. thesis, Universitat de Barcelona (2005)

  13. Quer-Sardanyons, L., Sanz-Solé, M.: Absolute continuity of the law of the solution to the 3-dimensional stochastic wave equation. J. Funct. Anal. 206(1), 1–32 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. Quer-Sardanyons, L., Sanz-Solé, M.: A stochastic wave equation in dimension 3: smoothness of the law. Bernoulli 10(1), 165–186 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  15. Sanz-Solé, M.: Malliavin calculus. With applications to stochastic partial differential equations. Fundamental Sciences. EPFL Press, Lausanne (2005)

    Google Scholar 

  16. Schwartz, D.: Théorie des distributions. Hermann, Paris (1966)

    MATH  Google Scholar 

  17. Walsh, J.B.: An introduction to stochastic partial differential equations. In: Hennequin, P.L. (ed.) École d’été de probabilités de Saint-Flour XIV – 1984. Lect. Notes Math. vol. 1180, pp. 265–437. Springer, Berlin (1986)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Lluís Quer-Sardanyons.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nualart, D., Quer-Sardanyons, L. Existence and Smoothness of the Density for Spatially Homogeneous SPDEs. Potential Anal 27, 281–299 (2007). https://doi.org/10.1007/s11118-007-9055-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11118-007-9055-3

Keywords

Mathematics Subject Classifications (2000)

Navigation