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Quenched to annealed transition in the parabolic Anderson problem
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  • Published: 08 August 2006

Quenched to annealed transition in the parabolic Anderson problem

  • M. Cranston1 &
  • S. Molchanov2 

Probability Theory and Related Fields volume 138, pages 177–193 (2007)Cite this article

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  • 18 Citations

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Abstract

We study limit behavior for sums of the form \(\frac{1}{|\Lambda_{L|}}\sum_{x\in \Lambda_{L}}u(t,x),\) where the field \(\Lambda_L=\left\{x\in {\bf{Z^d}}:|x|\le L\right\}\)is composed of solutions of the parabolic Anderson equation

$$u(t,x) = 1 + \kappa \mathop{\int}_{0}^{t} \Delta u(s,x){\rm d}s + \mathop{\int}_{0}^{t}u(s,x)\partial B_{x}(s). $$

The index set is a box in Z d, namely \(\Lambda_{L} = \left\{x\in {\bf Z}^{\bf d} : |x| \leq L\right\}\) and L = L(t) is a nondecreasing function \(L : [0,\infty)\rightarrow {\bf R}^{+}. \) We identify two critical parameters \(\eta(1) < \eta(2)\) such that for \(\gamma > \eta(1)\) and L(t) = eγ t, the sums \(\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)\) satisfy a law of large numbers, or put another way, they exhibit annealed behavior. For \(\gamma > \eta(2)\) and L(t) = eγ t, one has \(\sum_{x\in \Lambda_L}u(t,x)\) when properly normalized and centered satisfies a central limit theorem. For subexponential scales, that is when \(\lim_{t \rightarrow \infty} \frac{1}{t}\ln L(t) = 0,\) quenched asymptotics occur. That means \(\lim_{t\rightarrow \infty}\frac{1}{t}\ln\left (\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)\right) = \gamma(\kappa),\) where \(\gamma(\kappa)\) is the almost sure Lyapunov exponent, i.e. \(\lim_{t\rightarrow \infty}\frac{1}{t}\ln u(t,x)= \gamma(\kappa).\) We also examine the behavior of \(\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)\) for L = e γ t with γ in the transition range \((0,\eta(1))\)

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References

  1. Arous, B., Molchanov, R.: Transition from the annealed to the quenched asymptotics for a random walk on obstacles. Ann. Probab. (in press)

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

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  3. Cranston M., Molchanov S. (2005): Limit laws for sums of products of exponentials of iid random variables. Israel J. Math. 148, 115–136

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

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Califronia, Irvine, 281 MSTB, Irvine, CA, 92697-3875, USA

    M. Cranston

  2. Department of Mathematics, UNC-Charlotte, 376 Fretwell Bldg, Charlotte, NC, 28223-0001, USA

    S. Molchanov

Authors
  1. M. Cranston
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  2. S. Molchanov
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Corresponding author

Correspondence to M. Cranston.

Additional information

Research of first author supported by NSF grant DMS-0450756 of the second author by NSF grant DMS-0405927

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Cite this article

Cranston, M., Molchanov, S. Quenched to annealed transition in the parabolic Anderson problem. Probab. Theory Relat. Fields 138, 177–193 (2007). https://doi.org/10.1007/s00440-006-0020-7

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  • Received: 08 June 2005

  • Revised: 11 May 2006

  • Published: 08 August 2006

  • Issue Date: May 2007

  • DOI: https://doi.org/10.1007/s00440-006-0020-7

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Keywords

  • Parabolic Anderson model
  • Central limit theorem
  • Law of large numbers
  • Quenched asymptotics
  • Annealed asymptotics

Mathematics Subject Classifications (2000)

  • Primary 60F05
  • Primary 60F10
  • Secondary 60E07
  • Secondary 60G70
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