2010/10/07 by Schnitzler, Adrian, Wolff, Tilman · 1 citation
#60K37 #82C44 (Primary) 60H25 (Secondary) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1010.1512
We consider the solution u\colon [0,∞) ×ℤd→ [0,∞) to the parabolic Anderson model, where the potential is given by (t,x)↦γδYt(x) with Y a simple symmetric random walk on ℤd. 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,Yt+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.