PARABOLIC PROBLEMS FOR THE ANDERSON MODEL-II - 2ND-ORDER ASYMPTOTICS AND STRUCTURE OF HIGH PEAKS

This is a continuation of our previous work [6] on the investigation o f intermittency for the parabolic equation (partial derivative/partial derivative t)u = H u on IR+ x Z(d) associated with the Anderson Hamil tonian H = kappa Delta + xi(.) for i.i.d. random potentials xi(.). For the Cauchy problem with nonnegative homogeneous initial condition we study the second order asymptotics of the statistical moments [u(t, 0) (P)] and the almost sure growth of u(t, 0) as t --> infinity. We point out the crucial role of double exponential tails of xi(0) for the for mation of high intermittent peaks of the solution u(t, .) with asympto tically finite size. The challenging motivation is to achieve a better understanding of the geometric structure of such high exceedances whi ch in one or another sense provide the essential contribution to the s olution.