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On large deviations for the parabolic Anderson model
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  • Open Access
  • Published: 22 October 2009

On large deviations for the parabolic Anderson model

  • M. Cranston1,
  • D. Gauthier2 &
  • T. S. Mountford2 

Probability Theory and Related Fields volume 147, pages 349–378 (2010)Cite this article

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Abstract

The focus of this article is on the different behavior of large deviations of random functionals associated with the parabolic Anderson model above the mean versus large deviations below the mean. The functionals we treat are the solution u(x, t) to the spatially discrete parabolic Anderson model and a functional A n which is used in analyzing the a.s. Lyapunov exponent for u(x, t). Both satisfy a “law of large numbers”, with \({\lim_{t\to \infty} \frac{1}{t} \log u(x,t)=\lambda (\kappa)}\) and \({\lim_{n\to \infty} \frac{A_n}{n}=\alpha}\). We then think of αn and λ(κ)t as being the mean of the respective quantities A n and log u(t, x). Typically, the large deviations for such functionals exhibits a strong asymmetry; large deviations above the mean take on a different order of magnitude from large deviations below the mean. We develop robust techniques to quantify and explain the differences.

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References

  1. Cranston M., Gauthier D., Mountford T.: On large deviation regimes in random media models. Ann. Appl. Probab. 19(2), 826–862 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. Carmona R.A., Koralov L., Molchanov S.: Asymptotics for the almost sure Lyapunov exponent for the solution of the parabolic Anderson problem. Rand. Oper. Stoch. Equ. 9(1), 77–86 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Carmona R.A., Molchanov S.A.: Parabolic Anderson problem and intermittency. Mem. Am. Math. Soc. 108(518), viii+125 (1994)

    MathSciNet  Google Scholar 

  4. Cranston M., Mountford T., Shiga T.: Lyapunov exponents for the parabolic Anderson model. Acta Math. Univ. Comenian. 71(2), 163–188 (2002)

    MATH  MathSciNet  Google Scholar 

  5. Carmona R.A., Molchanov S.A., Viens F.: Sharp upper bound on the almost-sure exponential behavior of a stochastic partial equation. Rand. Oper. Stoch. Equ. 4(1), 43–49 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chow Y., Zhang Y.: Large deviations in first-passage percolation. Ann. Appl. Probab. 13(4), 1601–1614 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Durrett R.: Ten Lectures on Particle Systems, Ecole d’été de Probabilités de Saint Flour, XXIII. Springer, New York (1993)

    Google Scholar 

  8. Deuschel J.-D., Zeitouni O.: On increasing subsequences of I.I.D. samples. Combin. Probab. Comput. 8(3), 247–263 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Furuoya T., Shiga T.: Sample Lyapunov exponent for a class of linear Markovian systems over \({\mathbb Z^d}\). Osaka J. Math. 35, 35–72 (1998)

    MATH  MathSciNet  Google Scholar 

  10. Grimmett G.: Percolation. Springer, New York (1999)

    MATH  Google Scholar 

  11. Kesten, H.: First-passage percolation. From Classical to Modern Probability, Progr. Probab., vol. 54, pp. 93–143. Birkhäuser, Boston (2003)

  12. Kesten, H.: Aspects of first passage percolation. École d’été de probabilités de Saint-Flour. Lecture Notes in Math., vol. 1180, pp. 125–264. Springer, Heidelberg (1986)

  13. Liggett T.M.: An improved subadditive ergodic theorem. Ann. Probab. 13(4), 1279–1285 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  14. Molchanov, S., Ruzmaikin, A.: Lyapunov exponents and distributions of magnetic fields in dynamo models. The Dynkin Festschrift, pp. 287–306. Birkhäuser Boston, Boston (1994)

  15. Mountford T.S.: A note on limiting behaviour of disastrous environment exponents. Electron. J. Probab. 6(1), 9 (2001) (electronic)

    MathSciNet  Google Scholar 

  16. Revuz D., Yor M.: Continuous Martingales and Brownian Motion. Springer, Berlin (1998)

    Google Scholar 

  17. Shiga T.: Exponential decay rate of the survival probability in a disastrous random environment. Prob. Related Fields 108, 417–439 (1997)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Irvine, CA, 92612, USA

    M. Cranston

  2. DMA, EPFL, 1015, Lausanne, Switzerland

    D. Gauthier & T. S. Mountford

Authors
  1. M. Cranston
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  2. D. Gauthier
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  3. T. S. Mountford
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Corresponding author

Correspondence to M. Cranston.

Additional information

Research of the authors supported in part by a grant from NSF 0706198 and SFNS 3510767. D. Gauthier wishes to acknowledge the support of CNRS during his stay in Paris 2008.

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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Cranston, M., Gauthier, D. & Mountford, T.S. On large deviations for the parabolic Anderson model. Probab. Theory Relat. Fields 147, 349–378 (2010). https://doi.org/10.1007/s00440-009-0249-z

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  • Received: 03 November 2006

  • Revised: 10 February 2009

  • Published: 22 October 2009

  • Issue Date: May 2010

  • DOI: https://doi.org/10.1007/s00440-009-0249-z

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Keywords

  • Parabolic Anderson model
  • FKG inequality
  • Large deviations
  • Random media

Mathematics Subject Classification (2000)

  • Primary 60F10
  • 60K37
  • Secondary 60K35
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