WIAS Preprint No. 1669, (2011)

Parabolic Anderson model with finite number of moving catalysts



Authors

  • Castell, Fabienne
  • Gün, Onur
  • Maillard, Gregory

2010 Mathematics Subject Classification

  • 60H25 82C44 60F10 35B40

Keywords

  • Parabolic Anderson problem, catalytic random medium, intermittency, moment Lyapunov exponents

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, depicting how the total mass of $u$ concentrates as $ttoinfty$. Our results are both a generalization and an extension of the work of Gärtner and Heydenreich citegarhey06, where only the case $n=1$ was investigated.

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