Precise asymptotics for the parabolic Anderson model with a moving catalyst or trap

Abstract

We consider the solution ucolon [0,infty) timesmathbbZ^drightarrow [0,infty) to the parabolic Anderson model, where the potential is given by (t,x)mapstogammadelta_Y_tleft(xright) with Y a simple symmetric random walk on mathbbZ^d. Depending on the parameter gammain[-infty,infty), the potential is interpreted as a randomly moving catalyst or trap. In the trap case, i.e., gamma<0, we look at the annealed time asymptotics in terms of the first moment of u. Given a localized initial condition, we derive the asymptotic rate of decay to zero in dimensions 1 and 2 up to equivalence and characterize the limit in dimensions 3 and higher in terms of the Green's function of a random walk. For a homogeneous initial condition we give a characterisation of the limit in dimension 1 and show that the moments remain constant for all time in dimensions 2 and higher. In the case of a moving catalyst (gamma>0), we consider the solution u from the perspective of the catalyst, i.e., the expression u(t,Y_t+x). Focusing on the cases where moments grow exponentially fast (that is, gamma sufficiently large), we describe the moment asymptotics of the expression above up to equivalence. Here, it is crucial to prove the existence of a principal eigenfunction of the corresponding Hamilton operator. While this is well-established for the first moment, we have found an extension to higher moments.