Intermittency in a catalytic random medium

In this paper, we study intermittency for the parabolic Anderson equation ¿u/¿t=¿¿u+¿u, where u:Zd×[0,¿8)¿R, ¿ is the diffusion constant, ¿ is the discrete Laplacian and ¿:Zd×[0,¿8)¿R is a space-time random medium. We focus on the case where ¿ is ¿ times the random medium that is obtained by running independent simple random walks with diffusion constant ¿ starting from a Poisson random field with intensity ¿. Throughout the paper, we assume that ¿,¿¿,¿¿,¿¿¿(0,¿8). The solution of the equation describes the evolution of a "reactant" u under the influence of a "catalyst" ¿. We consider the annealed Lyapunov exponents, that is, the exponential growth rates of the successive moments of u, and show that they display an interesting dependence on the dimension d and on the parameters ¿,¿¿,¿¿,¿¿, with qualitatively different intermittency behavior in d=1,¿2, in d=3 and in d=4. Special attention is given to the asymptotics of these Lyapunov exponents for ¿¿0 and ¿¿8.