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Asymptotic Behavior of the Density in a Parabolic SPDE

Author

Listed:
  • A. Kohatsu-Higa

    (Universitat Pompeu Fabra)

  • D. Márquez-Carreras

    (Universitat de Barcelona)

  • M. Sanz-Solé

    (Universitat de Barcelona)

Abstract

Consider the density of the solution X(t, x) of a stochastic heat equation with small noise at a fixed t∈[0, T], x∈[0, 1]. In this paper we study the asymptotics of this density as the noise vanishes. A kind of Taylor expansion in powers of the noise parameter is obtained. The coefficients and the residue of the expansion are explicitly calculated. In order to obtain this result some type of exponential estimates of tail probabilities of the difference between the approximating process and the limit one is proved. Also a suitable iterative local integration by parts formula is developed.

Suggested Citation

  • A. Kohatsu-Higa & D. Márquez-Carreras & M. Sanz-Solé, 2001. "Asymptotic Behavior of the Density in a Parabolic SPDE," Journal of Theoretical Probability, Springer, vol. 14(2), pages 427-462, April.
    • Handle: RePEc:spr:jotpro:v:14:y:2001:i:2:d:10.1023_a:1011163714298
    DOI: 10.1023/A:1011163714298
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    References listed on IDEAS

    as
    1. Chenal, Fabien & Millet, Annie, 1997. "Uniform large deviations for parabolic SPDEs and applications," Stochastic Processes and their Applications, Elsevier, vol. 72(2), pages 161-186, December.
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