On the association and central limit theorem for solutions of the parabolic Anderson model

Abstract

We consider large scale behavior of the solution set $\{u(t,x): x\in{\mathbf{Z^d}}\}$ of the parabolic Anderson equation \begin{eqnarray*} u(t,x)&=&1+\kappa\int_0^t\Delta u(s,x)\, ds \\ &&{}+\int_0^tu(s,x)\,\partial W_x(s),\quad x\in{\mathbf{Z^d}},t\ge0, \end{eqnarray*} where $\{W_x : x\in{\mathbf{Z^d}}\}$ is a field of i.i.d. standard, one-dimensional Brownian motions, $\Delta$ is the discrete Laplacian and $\kappa>0.$ We establish that the properly normalized sum, $\sum_{x\in\Lambda _L}u(t,x),$ over spatially growing boxes $\Lambda_L=\{x\in{\mathbf {Z^d}}:\Vert x\Vert<L\}$ has an asymptotically normal distribution if the box $\Lambda_L$ grows sufficiently quickly with $t$ and provided $\kappa$ is sufficiently small depending on dimension. The asymptotic distribution of properly normalized sums over spatially growing disjoint boxes $\Lambda^1_L,\Lambda^2_L$ is asymptotically independent. Thus, on sufficiently large scales the field of solutions averaged over disjoint large boxes looks like an i.i.d. Gaussian field. We identify the variance of this Gaussian distribution in terms of the eigenfunction of the positive eigenvalue of the operator $2\kappa\Delta+\delta_0$.