Parabolic Anderson model with a finite number of moving catalysts

We consider the parabolic Anderson model (PAM) which is given by the equation ¿u=¿t = k¿u+¿u with u: Zd x[0;8)¿R, where k ¿ [0;8) is the diffusion constant, ¿ is the discrete Laplacian, and ¿: Zd x[0;8)¿R is a space-time random environment that drives the equation. The solution of this equation describes the evolution of a "reactant" u under the influence of a "catalyst" ¿. In the present paper we focus on the case where ¿ is a system of n independent simple random walks each with step rate 2d¿ 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. ¿ and show that these exponents, as a function of the diffusion constant k and the rate constant ¿, 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 ¿ 8. Our results are both a generalization and an extension of the work of G¨artner and Heydenreich [2], where only the case n = 1 was investigated.