Probability Theory and Related Fields

, Volume 138, Issue 1, pp 177–193

# Quenched to annealed transition in the parabolic Anderson problem

Article

DOI: 10.1007/s00440-006-0020-7

Cranston, M. & Molchanov, S. Probab. Theory Relat. Fields (2007) 138: 177. doi:10.1007/s00440-006-0020-7

## 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 modelCentral limit theoremLaw of large numbersQuenched asymptoticsAnnealed asymptotics

### Mathematics Subject Classifications (2000)

Primary 60F05Primary 60F10Secondary 60E07Secondary 60G70