Parabolic Anderson Model with a Finite Number of Moving Catalysts

Abstract

We consider the parabolic Anderson model (PAM) which is given by the equation ∂u∕∂t=κΔu+ξu with $u: \,{\mathbb{Z}}^{d} \times[0,\infty ) \rightarrow\mathbb{R}$, where κ∈[0,∞) is the diffusion constant, Δ is the discrete Laplacian, and $\xi : \,{\mathbb{Z}}^{d} \times[0,\infty ) \rightarrow\mathbb{R}$ is a space–time random environment. The solution of this equation describes the evolution of a “reactant” u under the influence of a “catalyst” ξ. In the present paper, we focus on the case where ξ is a system of n independent simple random walks each with step rate 2dρ and starting from the origin. We study the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of u w.r.t. ξ and show that these exponents, as a function of the diffusion constant κ and the rate constant ρ, behave differently depending on the dimension d. In particular, we give a description of the intermittent behavior of the system in terms of the annealed Lyapunov exponents, depicting how the total mass of u concentrates as t→∞. Our results are both a generalization and an extension of the work of Gärtner and Heydenreich [3], where only the case n=1 was investigated.