Quenched to annealed transition in the parabolic Anderson problem

We study limit behavior for sums of the form $frac{1}{|Lambda_{L|}}sum_{xin Lambda_{L}}u(t,x),$ where the field $Lambda_L=left{xin {bf{Z^d}}:|x|le Lright}$ 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 $Lambda_{L} = left{xin {bf Z}^{bf d} : |x| leq Lright}$ 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) = e^{γ t} , the sums $frac{1}{|Lambda_L|}sum_{xin Lambda_L}u(t,x)$ satisfy a law of large numbers, or put another way, they exhibit annealed behavior. For $gamma > eta(2)$ and L(t) = e^{γ t} , one has $sum_{xin Lambda_L}u(t,x)$ 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 $lim_{trightarrow infty}frac{1}{t}lnleft (frac{1}{|Lambda_L|}sum_{xin Lambda_L}u(t,x)right) = gamma(kappa),$ where $gamma(kappa)$ is the almost sure Lyapunov exponent, i.e. $lim_{trightarrow infty}frac{1}{t}ln u(t,x)= gamma(kappa).$ We also examine the behavior of $frac{1}{|Lambda_L|}sum_{xin Lambda_L}u(t,x)$ for L = e ^{γ t} with γ in the transition range $(0,eta(1))$