Annealed asymptotics for the parabolic Anderson model with a moving catalyst

This paper deals with the solution uu to the parabolic Anderson equation ∂u/∂t=κΔu+ξu∂u/∂t=κΔu+ξu on the lattice ZdZd. We consider the case where the potential ξξ is time-dependent and has the form ξ(t,x)=δ0(x−Yt)ξ(t,x)=δ0(x−Yt) with YtYt being a simple random walk with jump rate 2dϱ2dϱ. The solution uu may be interpreted as the concentration of a reactant under the influence of a single catalyst   particle YtYt.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 κκ and ϱϱ.