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
Cranston M., Gauthier D., Mountford T.: On large deviation regimes in random media models. Ann. Appl. Probab. 19(2), 826–862 (2009)
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)
Carmona R.A., Molchanov S.A.: Parabolic Anderson problem and intermittency. Mem. Am. Math. Soc. 108(518), viii+125 (1994)
Cranston M., Mountford T., Shiga T.: Lyapunov exponents for the parabolic Anderson model. Acta Math. Univ. Comenian. 71(2), 163–188 (2002)
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)
Chow Y., Zhang Y.: Large deviations in first-passage percolation. Ann. Appl. Probab. 13(4), 1601–1614 (2003)
Durrett R.: Ten Lectures on Particle Systems, Ecole d’été de Probabilités de Saint Flour, XXIII. Springer, New York (1993)
Deuschel J.-D., Zeitouni O.: On increasing subsequences of I.I.D. samples. Combin. Probab. Comput. 8(3), 247–263 (1999)
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)
Grimmett G.: Percolation. Springer, New York (1999)
Kesten, H.: First-passage percolation. From Classical to Modern Probability, Progr. Probab., vol. 54, pp. 93–143. Birkhäuser, Boston (2003)
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)
Liggett T.M.: An improved subadditive ergodic theorem. Ann. Probab. 13(4), 1279–1285 (1985)
Molchanov, S., Ruzmaikin, A.: Lyapunov exponents and distributions of magnetic fields in dynamo models. The Dynkin Festschrift, pp. 287–306. Birkhäuser Boston, Boston (1994)
Mountford T.S.: A note on limiting behaviour of disastrous environment exponents. Electron. J. Probab. 6(1), 9 (2001) (electronic)
Revuz D., Yor M.: Continuous Martingales and Brownian Motion. Springer, Berlin (1998)
Shiga T.: Exponential decay rate of the survival probability in a disastrous random environment. Prob. Related Fields 108, 417–439 (1997)
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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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DOI: https://doi.org/10.1007/s00440-009-0249-z