Parabolic Anderson model with a finite number of moving catalysts

TL;DR

This paper analyzes the parabolic Anderson model with multiple moving catalysts, focusing on the growth rates of solutions influenced by random walk catalysts, revealing dimension-dependent intermittent behavior and extending previous single-catalyst results.

Contribution

It generalizes prior work by studying multiple catalysts and characterizes the intermittent behavior through annealed Lyapunov exponents across different dimensions.

Findings

Lyapunov exponents depend on dimension and parameters

Intermittent mass concentration occurs as time grows

Results extend single-catalyst analysis to multiple catalysts

Abstract

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.