, Volume 138, Issue 1-2, pp 177-193
Date: 08 Aug 2006

Quenched to annealed transition in the parabolic Anderson problem

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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 \((0,\eta(1))\)

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