Journal of Statistical Physics

, Volume 78, Issue 5–6, pp 1377–1401

The stochastic heat equation: Feynman-Kac formula and intermittence

  • Lorenzo Bertini
  • Nicoletta Cancrini
Articles

Abstract

We study, in one space dimension, the heat equation with a random potential that is a white noise in space and time. This equation is a linearized model for the evolution of a scalar field in a space-time-dependent random medium. It has also been related to the distribution of two-dimensional directed polymers in a random environment, to the KPZ model of growing interfaces, and to the Burgers equation with conservative noise. We show how the solution can be expressed via a generalized Feynman-Kac formula. We then investigate the statistical properties: the two-point correlation function is explicitly computed and the intermittence of the solution is proven. This analysis is carried out showing how the statistical moments can be expressed through local times of independent Brownian motions.

Key Words

Stochastic partial differential equations Feynman-Kac formula random media moment Lyapunov exponents intermittence local times 

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References

  1. 1.
    P. H. Baxendale and B. L. Rozowskii, Kinematic dynamo and intermittence in a turbulent flow, CAMS preprint #92-2 (1992).Google Scholar
  2. 2.
    L. Bertini, N. Cancrini, and G. Jona-Lasinio, The stochastic Burgers equation,Commun. Math. Phys. 165:211–232 (1994).Google Scholar
  3. 3.
    L. Bertini, N. Cancrini, and G. Jona-Lasinio, Burgers equation forced by conservative or nonconservative noise,Proceedings of the NATO-ASI School “Stochastic Analysis and Applications in Physics” (Madeira, August 1993), to appear.Google Scholar
  4. 4.
    L. V. Bogachev, The moment approach to intermittence of random media. The model of a mean-field diffusion, Preprint Nr. 116, Institut für Mathematik Ruhr-Universität-Bochum (1991).Google Scholar
  5. 5.
    N. Cancrini, Soluzione del problema di Cauchy per l'equazione di Burgers stocastica in una dimensione spaziale, Ph.D. dissertation, Dipartimento di Fisica, Università di Roma “La Sapienza” (1993) [in Italian].Google Scholar
  6. 6.
    G. Da Prato, A. Debussche, and R. Temam, Stochastic Burger equation, Preprint di Matematica n. 27, Scuola Normale Superiore, Pisa (1993).Google Scholar
  7. 7.
    I. I. Gikhman and A. V. Skorohod,Stochastic Differential Equations (Springer-Verlag, Berlin, 1972).Google Scholar
  8. 8.
    H. Holden, T. Lindstrøm, B. Øksendal, J. Ubøe, and T.-S. Zhang, The Burgers equation with a noisy force and the stochastic heat equation,Commun. Partial Differential Equations 19:119–141 (1994).Google Scholar
  9. 9.
    M. Kardar, Replica Bethe ansatz studies of two-dimensional interfaces with quenched random impurities,Nucl. Phys. B 290:582–602 (1987).Google Scholar
  10. 10.
    M. Kardar, G. Parisi, and Y.-C. Zhang, Dynamical scaling of growing interfaces,Phys. Rev. Lett. 56:889–892 (1986).Google Scholar
  11. 11.
    M. Kardar and Y.-C. Zhang, Scaling of directed polymers in random media,Phys. Rev. Lett. 58:2087–2090 (1987).Google Scholar
  12. 12.
    J. Krug and H. Spohn, Kinetic roughening of growing surfaces, inSolids far from equilibrium: Growth, Morphology and Defects, C. Godrèche, ed. (Cambridge University Press, Cambridge, 1991).Google Scholar
  13. 13.
    S. A. Molchanov, Ideas in theory of random media,Acta Appl. Math. 22:139–282 (1991), and references therein.Google Scholar
  14. 14.
    C. Mueller, On the support of solutions to the heat equation with noise,Stochastics Stochastics Rep. 37:225–245 (1991).Google Scholar
  15. 15.
    J. M. Noble, Evolution equation with Gaussian potential, Preprint (1993).Google Scholar
  16. 16.
    L. Pastur, Large time behaviour of moments of solution of parabolic differential equations with random coefficients, Séminaire 1991/92 Centre de Mathematiques, Ecole Polytechnique, Palaiseau, France.Google Scholar
  17. 17.
    D. Revuz and M. Yor,Continuous Martingales and Brownian Motion (Springer-Verlag, Berlin, 1991).Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • Lorenzo Bertini
    • 1
  • Nicoletta Cancrini
    • 3
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew York
  2. 2.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomeItaly
  3. 3.Dipartimento di FisicaUniversità di Roma “La Sapienza”RomeItaly

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