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

TL;DR

This paper derives precise asymptotic behaviors for the parabolic Anderson model with a moving catalyst or trap, analyzing how the solution's moments evolve over time in various dimensions and initial conditions.

Contribution

It provides new asymptotic results for the model with moving catalysts or traps, including the existence of principal eigenfunctions for higher moments.

Findings

In dimensions 1 and 2, the solution decays to zero with specific rates.

In dimensions 3 and higher, the limit behavior is characterized via Green's functions.

For large $\gamma$, moments grow exponentially fast, with new eigenfunction results for higher moments.

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.