WIAS Preprint No. 1553, (2010)

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



Authors

  • Schnitzler, Adrian
  • Wolff, Tilman

2010 Mathematics Subject Classification

  • 60K37 82C44 60H25

Keywords

  • parabolic Anderson model, annealed asymptotics, dynamic random medium

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.

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