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

Abstract

We consider the solution $u\colon [0,\infty) \times\mathbb{Z}^d\rightarrow [0,\infty) $ to the parabolic Anderson model, where the potential is given by $(t,x)\mapsto\gamma\delta_{Y_t}(x)$ with $Y$ a simple symmetric random walk on $\mathbb{Z}^d$. Depending on the parameter $\gamma\in[-\infty,\infty)$, 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.