On the association and Central Limit Theorem for solutions of the Parabolic Anderson Model
Monday, January 14, 2013 - 2:00pm - 2:50pm
We consider large scale behavior of the solution set of values u(t,x) for x in the d-dimensional integer lattice of the parabolic Anderson equation. We establish that the properly normalized sums of the u(t,x), over spatially growing boxes have an asymptotically normal distribution if the box grows sufficiently quickly with t and provided intermittency holds. The asymptotic distribution of properly normalized sums over spatially growing disjoint boxes is asymptotically independent. Thus, on sufficiently large scales the field of solutions averaged over disjoint large boxes looks like an iid Gaussian field. We identify the variance of this Gaussian distribution in terms of the eigenfunction of the positive eigenvalue of a certain operator.