Probability Theory and Related Fields

, Volume 138, Issue 1–2, pp 177–193 | Cite as

Quenched to annealed transition in the parabolic Anderson problem

Article

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 Zd, 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 = eγ t with γ in the transition range \((0,\eta(1))\)

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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References

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Califronia, IrvineIrvineUSA
  2. 2.Department of MathematicsUNC-CharlotteCharlotteUSA

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