Journal of Theoretical Probability

, Volume 9, Issue 4, pp 863–901 | Cite as

The law of the solution to a nonlinear hyperbolicSPDE

  • Carles Rovira
  • Marta Sanz-Solé


Let {\(\dot W\) s,t ,(s,t∈ℝ + 2 } be a white noise on ℝ + 2 . We consider the hyperbolic stochastic partial differential equation {ie863-3} The purpose of this paper is to study the law of the solution to this equation. We analyze the existence and smoothness of the density using the tools of Malliavin Calculus. Finally we prove a large deviation principle on the space of continuous functions, for the family of probabilities obtained by perturbation of the noise in the equation.

Key Words

HyperbolicSPDE's strong and weak solution Malliavin calculus large deviations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Azencott, R. (1980). Grandes déviations et applications; École d'été de Probabilités de Saint-Flour VIII, Springer Verlag,Lect. Notes. Math. 779, 1–176.Google Scholar
  2. 2.
    Bally, V., and Pardoux, E. (1993). Malliavin calculus for white noise driven parabolic spde's. Preprint.Google Scholar
  3. 3.
    Cairoli, R. (1972). Sur une équation différentielle stochastique.C.R. Acad. Sci. Paris 274, 1739–1742.Google Scholar
  4. 4.
    Cairoli, R., and Walsh, J. B. (1975). Stochastic integrals in the plane.Acta Math. 134, 111–183.Google Scholar
  5. 5.
    Doss, H., and Dozzi, M. (1986). Estimations des grandes déviations pour les processus de diffusion à paramètre multidimensionnel.Séminaire de Probabilités XX. Springer Verlag,Lect. Notes Math. 1204, 68–79.Google Scholar
  6. 6.
    Doss, H., and Priouret, P. (1983). Petites perturbations des systèmes dynamiques avec reflexion. Springer Verlag.Séminaire de Probabilités XVII. Lect. Notes Math. 986, 353–370.Google Scholar
  7. 7.
    Farré, M. (1995). Hyperbolic stochastic differential equations: absolute continuity of the law of the solution at a fixed point.Appl. Math. Optim. (to appear).Google Scholar
  8. 8.
    Farré, M., and Nualart, D. (1993). Nonlinear stochastic integral equation in the plane.Stoch. Proc. Appl. 46, 219–239.Google Scholar
  9. 9.
    Guyon, X., and Prum, B. (1980). Semimartingales à indices dans ℝ+2. Thèse de Doctorat d'État. Sci. Math. de Paris Sud.Google Scholar
  10. 10.
    Ikeda, N., and Watanabe, S. (1981).Stochastic Differential Equations and Diffusion Processes. North Holland, Tokyo.Google Scholar
  11. 11.
    Lépingle, D., Nualart, D., and Sanz-Solé, M. (1989). Dérivation stochastique des diffusions réflechies.Annals de l'Institut H. Poincaré 25(3), 283–306.Google Scholar
  12. 12.
    Léandre, R., and Russo, F. (1990). Estimation de Varadhan pour des diffusions à deux paramètres.Prob. Th. Rel. Fields 84, 429–451.Google Scholar
  13. 13.
    Liptser, R. S., and Shiryayev, A. N. (1977).Statistics of Random Processes I. General Theory. Springer Verlag.Google Scholar
  14. 14.
    Malliavin, P. (1978). Stochastic calculus of variations and hypoelliptic operators.Proc. Int'l. Conf. Stoch. Differ. Eqs. Kyoto 1976, pp. 195–263. John Wiley.Google Scholar
  15. 15.
    Millet, A., Nualart, D., and Sanz-Solé, M. (1992). Large deviations for a class of anticipating stochastic differential equations.Ann. Prob. 20, 1902–1931.Google Scholar
  16. 16.
    Norris, J. R. (1995). Twisted Sheets.J. Funct. Analysis 132, 273–334.Google Scholar
  17. 17.
    Nualart, D., and Sanz-Solé, M. (1985). Malliavin calculus for two-parameter Wiener functionals.Z. Wahrsch. Verw. Gebiete 70, 573–590.Google Scholar
  18. 18.
    Nualart, D., and Sanz-Solé, M. (1989). Stochastic Differential Equtions on the plane: Smoothness of the solution.J. Multivariate Anal. 31, 1–29.Google Scholar
  19. 19.
    Rovira, C., and Sanz-Solé, M. (1995). A nonlinear hyperbolic SPDE: Approximations and support. In Etheridge, A. (ed.),Stochastic Partial Differential Equations pp. 241–261.London Mathematical Society Lecture Note Series 216. Cambridge University Press.Google Scholar
  20. 20.
    Sowers, R. B. (1992). Large deviations for a reaction-diffusion equation with non-Gaussian perturbations.Ann. Prob. 20, 504–537.Google Scholar
  21. 21.
    Wong, E., and Zakai, M. (1974). Weak martingales and stochastic integrals in the plane.Ann. Prob. 4, 570–586.Google Scholar
  22. 22.
    Yeh, H. (1981). Existence of strong solutions for stochastic differential equations in the plane.Pacific J. Math. 97, 217–297.Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • Carles Rovira
    • 1
  • Marta Sanz-Solé
    • 1
  1. 1.Facultat de MatemàtiquesUniversitat de BarcelonaBarcelonaSpain

Personalised recommendations