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Relating the Almost-Sure Lyapunov Exponent of a Parabolic SPDE and its Coefficients’ Spatial Regularity

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Abstract

We derive a lower bound on the large-time exponential behavior of the solution to a stochastic parabolic partial differential equation on R+×R in the case of a spatially homogeneous Gaussian potential that is white-noise in time, and study the relation between the lower bound and the potential’s spatial modulus of continuity.

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References

  1. Adler, R.: An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes, Inst. Math. Stat., Hayward, CA, 1990.

    Google Scholar 

  2. Berge, B., Chueshov, I.D. and Vuillermot, P.-A.: ‘Lyapunov exponents and stability for nonlinear SPDE’s driven by finite-dimensional Wiener processes’, C.R. Acad. Sci. Paris Sér. I Math. 329(3) (1999), 215–220.

    Google Scholar 

  3. Bertini, L. and Giacomin, G.: ‘On the long-time behavior of the stochastic heat equation’, Probab. Theory Related Fields 114(3) (1999), 279–289.

    Google Scholar 

  4. Carmona, R.A. and Molchanov, S.A.: Parabolic Anderson Model and Intermittency, Mem. Amer. Math. Soc. 418, 1994.

  5. Carmona, R.A., Molchanov, S.A. and Viens, F.G.: ‘Sharp upper bound on exponential behavior of a stochastic partial differential equation’, Random Operators and Stochastic Equations 4(1) (1996), 43–49.

    Google Scholar 

  6. Carmona, R. and Viens, F.: ‘Almost-sure exponential behavior of a stochastic Anderson model with continuous space parameter’, Stochastics Stochastics Rep. 62(3–4) (1998), 251–273.

    Google Scholar 

  7. Cranston, M., Mountford, T.S. and Shiga, T.: ‘Lyapunov exponents for the parabolic Anderson model’, Acta Math. Univ. Comenian. (N.S.) 71(2) (2002), 163–188.

    Google Scholar 

  8. Da Prato, G. and Zabczyk, J.: Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, 1992.

  9. Dalang, R. and Frangos, N.: ‘The stochastic wave equation in two spatial dimensions’, Ann. Probab. 26 (1998), 187–212.

    Google Scholar 

  10. Gartner, J., Konig, W. and Molchanov, S.A.: ‘Almost sure asymptotics for the continuous parabolic Anderson model’, Probab. Theory Related Fields 118(4) (2000), 547–573.

    Google Scholar 

  11. Karatzas, I. and Shreve, S.: Brownian Motion and Stochastic Calculus, 2nd edn, Springer, 1991.

  12. Millet, A. and Sanz-Sole, M.: ‘A stochastic wave equation in two spatial dimensions: Smoothness of the law’, Ann. Probab. 27(2) (1999), 803–844.

    Google Scholar 

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

    Google Scholar 

  14. Tindel, S. and Viens, F.: ‘Almost sure exponential behaviour for a parabolic SPDE on a manifold’, Stochastic Process. Appl. 100 (2002), 53–74.

    Google Scholar 

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Authors and Affiliations

  1. Institut Élie Cartan, Université Henri Poincaré (Nancy), BP 239, 54506, Vandoeuvre-lès-Nancy, France

    Samy Tindel

  2. Dept. Mathematics & Dept. Statistics, Purdue University, 150 N. University St., West Lafayette, IN, 47907-2067, USA

    Frederi Viens

Authors
  1. Samy Tindel
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  2. Frederi Viens
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Correspondence to Samy Tindel.

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Mathematics Subject Classifications (2000)

60H15, 60G15.

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Tindel, S., Viens, F. Relating the Almost-Sure Lyapunov Exponent of a Parabolic SPDE and its Coefficients’ Spatial Regularity. Potential Anal 22, 101–125 (2005). https://doi.org/10.1007/s11118-004-0576-8

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  • DOI: https://doi.org/10.1007/s11118-004-0576-8

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