Asymptotics for the Boundary Parabolic Anderson Problem in a Half Space∗†

We study the large time behavior of the solutions of the Cauchy problem for the Anderson model restricted to the upper half space D = Zd−1 × Z+ and/or D = Rd−1 × R+ when the potential is a homogeneous random field concentrated on the boundary ∂D. In other words we consider the problem: ∂u(t, z) ∂t = κ∆u(t, z) + ξ(x)u(t, z) z = (x, y), with y ≥ 0 and t ≥ 0 with an appropriate initial condition. We determine the large time asymptotics of the moments of the solutions as well as their almost sure asymptotic behavior when t→∞ and when the distance from the boundary, i.e. y = y(t) goes simultaneously to infinity as a function of the time t. We identify the rates of escape of y(t) which correspond to specific behaviors of the solutions and different types of dependence upon the diffusivity constant κ. We also show that the case of the lattice differs drastically from the continuous case when it comes to the existence of the moments and the influence of κ. Intermittency is proved as a consequence of the large time behavior of the solutions. ∗Running Title: Anderson Problem in a Half Space †AMS Subject Classification: 60H25 ‡Partially supported by ONR N00014-91-1010