Parabolic Anderson model with a finite number of moving catalysts

Details Parabolic Anderson model with a finite number of moving catalysts Parabolic Anderson model with a finite number of moving catalysts

2010-10-23

arxiv.org/abs/1010.4868v2

Mathematics

Probability

In honour of J\"urgen G\"artner on the occasion of his 60th birthday, 25 pages. Updated version following the referee's commen

Scientific paper

We consider the parabolic Anderson model (PAM) which is given by the equation $\partial u/\partial t = \kappa\Delta u + \xi u$ with $u\colon\, \Z^d\times [0,\infty)\to \R$, where $\kappa \in [0,\infty)$ is the diffusion constant, $\Delta$ is the discrete Laplacian, and $\xi\colon\,\Z^d\times [0,\infty)\to\R$ 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" $\xi$. In the present paper we focus on the case where $\xi$ is a system of $n$ independent simple random walks each with step rate $2d\rho$ and starting from the origin. We study the \emph{annealed} 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 $t\to\infty$. Our results are both a generalization and an extension of the work of G\"artner and Heydenreich 2006, where only the case $n=1$ was investigated.

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