Asymptotics for the boundary parabolic Anderson problem in a half space

Abstract

We study the large time behavior of the solutions of the Cauchy problem for the Anderson model restricted to the upper half space and/or when the potential is a homogeneous random field concentrated on the boundary . In other words we consider the problem: 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.