Precise Asymptotics for the Parabolic Anderson Model with a Moving Catalyst or Trap

Abstract

We consider the solution $u: [0,\infty ) \times{\mathbb{Z}}^{d} \rightarrow[0,\infty )$ to the parabolic Anderson model, where the potential is given by $(t,x)\mapsto \gamma {\delta }_{{Y }_{t}}\left (x\right )$ with Y a simple symmetric random walk on ${\mathbb{Z}}^{d}$. Depending on the parameter γ∈[−∞,∞), the potential is interpreted as a randomly moving catalyst or trap.

In the trap case, i.e., γ<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 (γ>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, γ 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.

AMS 2010 Subject Classification. Primary 60K37, 82C44; Secondary 60H25.