Parabolic Anderson model with a finite number of moving catalysts

Abstract

We consider the parabolic Anderson model (PAM) which is given by the equation partial u/partial t = kappaDelta u + xi u with ucolon, Z^dtimes [0,infty)to R, where kappa in [0,infty) is the diffusion constant, Delta is the discrete Laplacian, and xicolon,Z^dtimes [0,infty)toR is a space-time random environment. The solution of this equation describes the evolution of the density u of a ``reactant'' u under the influence of a ``catalyst'' xi.newlineindent In the present paper we focus on the case where xi is a system of n independent simple random walks each with step rate 2drho and starting from the origin. We study the emphannealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of u w.r.t. xi and show that these exponents, as a function of the diffusion constant kappa and the rate constant rho, 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,...