A Generalized Feynman-Kac Formula for the Stochastic Heat Problem with Anticipating Initial Conditions

  • Fred Espen Benth
Conference paper
Part of the Progress in Probability book series (PRPR, volume 38)


Using White Noise Analysis, we construct a Feynman-Kac formula for the stochastic heat equation with anticipating initial conditions. The obtained solution has applications to nonlinear filtering and heat transport with noise.


Infinitesimal Generator Stochastic Partial Differential Equation Coordinate Process Heat Problem Stochastic Heat Equation 
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  1. [B1]
    F. E. Benth: Integrals In The Hida Distribution Space (S)*. In T. Lindstrøm, B. Øksendal and A. S. Ustunel (eds.): Stochastic Analysis and Related Topics, Stochastic Monographs Vol 8. Gordon and Breach Science Publishers. 1993.Google Scholar
  2. B2] F. E. Benth: A Functional Process Solution to a Stochastic Partial Differential Equation with Applications to Nonlinear Filtering. To appear in Stochastics.Google Scholar
  3. [B3]
    F. E. Benth: On the Positivity of the Stochastic Heat Equation. Manuscript, University of Mannheim, 1994.Google Scholar
  4. [BC]
    L. Bertini and N. Cancrini: The Stochastic heat equation and Intermittence. Preprint no. 1032, University of Rome (La Sapienza), 1994.Google Scholar
  5. [GK]
    I. Gyöngy & N. V. Krylov: SPDEs With Unbounded Coefficients I & II; Stochastics 32–33, 1990.Google Scholar
  6. [HLØUZ]
    H. Holden, T. Lindstrøm, B. Øksendal, J. Ubøe and T. -S. Zhang: The Burgers Equation with a Noisy Force. Comm. PDE. 19, 1994.Google Scholar
  7. [HKPS]
    T. Hida, H.-H. Kuo, J. Potthoff and L. Streit: White Noise -An Infinite Dimensional Calculus. Kluwer Academic Press, 1993.MATHGoogle Scholar
  8. [K]
    H.-H. Kuo: Convolution and Fourier Transform of Hida Distributions. Manuscript, Lousiana State University, USA, 1990.Google Scholar
  9. [KLPSW]
    Yu. G. Kondratiev, P. Leukert. J. Potthoff, L. Streit, W. Westerkamp: Generalized Functionals in Gaussian Spaces - The Characterization Theorem Revisited. Preprint, University of Mannheim, 1994.Google Scholar
  10. [KS]
    I. Karatzas & S. Shreve: Brownian Motion and Stochastic Calculus. Springer 1988.MATHGoogle Scholar
  11. [KR]
    N. V. Krylov and B. L. Rozovskii: Stochastic Partial Differential Equations and Diffusions Processes. Russian Math. Surveys. 37, 1982.Google Scholar
  12. [LØU1]
    T. Lindstrøm, B. Øksendal and J. Ubøe: Stochastic Modelling of Fluid Flow in a Porous Medium. In S. Chen and J. Yong (eds.): Control Theory, Stochastic Analysis and Applications. World Scientific. 1991.Google Scholar
  13. [LØU2]
    T. Lindstrøm, B. Øksendal and J. Ubøe: Wick Multiplication and Ito- Skorohod Stochastic Differential Equations. In S. Albeverio et. al. (eds.): Ideas and Methods in Mathematical Analysis, Stochastics and Applications. Cambridge University Press. 1992.Google Scholar
  14. LR] R. Léandre and F. Russo: Estimation de la Densité de la Solution de l’Équation de Zakai Robuste. To appear in Journal of Potential Analysis.Google Scholar
  15. [NZ]
    D. Nualart and M. Zakai: Generalized Stochastic Integrals and the Malliavin Calculus. Prob. Th. Rel. Fields 73, 1986.Google Scholar
  16. [Pa]
    E. Pardoux: Stochastic Partial Differential Equations and Filtering of Diffusion Processes. Stochastics 3, 1979.Google Scholar
  17. [P]
    J. Potthoff: White Noise Approach to Parabolic Stochastic Partial Differential Equations. Preprint, University of Mannheim, 1994.Google Scholar
  18. [PS]
    J. Potthoff and L. Streit: A Characterization of Hida Distributions. J. Funct. Anal. 101, 1991.Google Scholar
  19. TB] M. Timpel and F. E. Benth: Topological Aspects of the Characterization of Hida Distributions - A Remark. To appear in Stochastics.Google Scholar
  20. [Z]
    M. Zakai: On the Optimal Filtering of Diffusion Processes. Z. Wahrschein, verw. Geb.. 11, 1969.Google Scholar

Copyright information

© Birkhäuser Boston 1996

Authors and Affiliations

  • Fred Espen Benth
    • 1
    • 2
  1. 1.University of OsloOsloNorway
  2. 2.University of Mannheim A5MannheimGermany

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