Annealed asymptotics for the parabolic Anderson model with a moving catalyst

This paper deals with the solution u to the parabolic Anderson equation partial derivative u/partial derivative t = kappa Delta u+xi u on the lattice Z(d). We consider the case where the potential is time-dependent and has the form xi(t, x) = delta(0)(x - Y-t) with Y-t being a simple random walk with jump rate 2d rho. The solution u may be interpreted as the concentration of a reactant under the influence of a single catalyst particle Y-t. In the first part of the paper we show that the moment Lyapunov exponents coincide with the upper boundary of the spectrum of certain Hamiltonians. In the second part we study intermittency in terms of the moment Lyapunov exponents as a function of the model parameters kappa and rho. (C) 2006 Elsevier B.V. All rights reserved.